I’m back on Twitter, and you can find me here: https://twitter.com/charlesdavihat1

I regret to say that the quality of national discourse has declined notably in my absence…

I’m back on Twitter, and you can find me here: https://twitter.com/charlesdavihat1

I regret to say that the quality of national discourse has declined notably in my absence…

Advertisements

Imagine we had a line segment with a finite length, and then iteratively extracted the center ; and then extracted the center of the two resulting segments, and so on.

The set of points that remains after this process is of course the Cantor Set:

https://en.wikipedia.org/wiki/Cantor_set

Note that this set will be bounded by the two end points of the original line segment.

Now imagine that we gave the line segment some velocity, sending the entire Cantor Set in between, in tact, through space. Further, imagine that we had a sensor at a fixed point along the path of the set that lights up every time a point in the set crosses the sensor.

Because there are a countable number of gaps in the line segment, the light will blink on and off with some frequency, an infinite number of times. The signal generated will depend upon both the gaps in the set, and the velocity of the line segment.

Also note that the amount of time it takes for the line segment to cross the sensor is given by , where is the length of the segment, and is the velocity of the segment. Because and are both finite, is finite.

Now imagine that we have two such line segments and , both of length , but that is travelling with a faster velocity of . Because , it will take less time for to cross the sensor, causing the sensor to be triggered for a shorter amount of time by than .

For example, the length of the gap in the middle (the largest gap initially removed) has a length of . The amount of time it takes for this gap to cross the sensor is , which will obviously depend upon the velocity of the segment. If we assume that the light turns off once the sensor hits this gap, then the amount of time the light is off during this gap will vary with the velocity of the segment.

The same will be true of all gaps in the set.

This implies that the signal generated by is objectively distinguishable from the signal generated by , despite the fact that both will cause the sensor to trigger an infinite number of times.

This same thought experiment works with any bounded set that has a countable number of “holes” in it.

Note that this (admittedly theoretical) hypothetical suggests that an infinite signal can be conveyed in a finite amount of time.

In this article, I apply my model of AI to image classification. The working paper is available on researchgate, and is also attached as a pdf.

A New Model of Artificial Intelligence: Application to Data II

In this note, I’ll present an unsupervised algorithm that can extract three-dimensional features from an ordinary two-dimensional image, and detect edges within the image, thereby extracting two-dimensional shape information, in each case in polynomial time.

A research note explaining the algorithm, together with the code, is available on my researchgate homepage here, and I’ve also attached a PDF.

The relevant scripts are also attached as PDFs below. Any missing scripts can be found in previous posts, and in the “Algorithms” tab.

In a previous post, I showed how information theory can be used to explain why it is that human beings perceive changes in luminosity logarithmically, and not linearly, as a function of actual luminosity (see, “Using Information Theory to Explain Color Perception“). In short, I showed that if you assume that what human beings perceive as luminosity is actually a measure of information content, then perceived luminosity should be a logarithmic function of actual physical luminosity, which is indeed the case (see, for example, Fechner’s Law). In this post, I’ll show how we can use similar concepts to explain why it is that human beings perceive the colors near the boundary of the green and blue portion of the spectrum as brighter than the colors near the red portion of the spectrum.

**Luminosity, Energy, Information, and Perception**

In a very long paper posted below (that you absolutely do not need to read to understand what follows), I showed that assuming that light literally contains information has a lot of interesting consequences, and even implies that time-dilation will occur. In this post, I’ll take a slightly different approach that nonetheless assumes that the environment with which we interact can be thought of as literally containing information. This doesn’t require us to assume that we’re living in The Matrix, but rather, that at a fundamental level, all of our perceptions require representations of external stimuli to be generated, which means that our bodies are quite literally generating information. Because information is measured in bits, this view implies that we can measure the information content of external stimuli in bits.

Specifically, we can measure the sensory information triggered by a single photon landing on a human eye by making use of some basic principles from contemporary physics. For those that are unfamiliar with Planck’s equation, , in short, it gives us the energy of a single “parcel” of light , known as a photon, as a function of its frequency . The constant is known as Planck’s constant, given that Max Planck penned the equation, though Einstein also had a role in popularizing the view that light is “quantized” into photons through his work on the photoelectric effect.

Returning to our analysis, as noted above, human beings perceive changes in luminosity logarithmically as a function of actual physical luminosity. So, for example, if a light source has a luminosity of , then the level of luminosity perceived by a human observer of that source is given by . We showed below that we can explain this phenomenon, as well as other related phenomena, by assuming that what is perceived as luminosity is actually a representation of the amount of information observed. That is, the perceived brightness of a light source is not a direct representation of the luminosity of the light, but is instead a representation of the amount of information required to represent the luminosity of the source. In crude (and imprecise) terms, if the luminosity of a light source is 10, then we need bits to represent its luminosity, meaning that what we perceive as brightness should actually be measured in bits.

In more precise terms, luminosity is also a measure of energy, but it is not a measure of the energy of the individual photons ejected by a light source (like Planck’s equation). Luminosity is instead a measure of the total energy ejected by a light source. As a result, when we take the logarithm of the luminosity of a given light source, we are taking the logarithm of an amount of energy. Generalizing upon this, it should also be the case that the energy of an individual photon incident upon a human eye is also perceived logarithmically. As it turns out, assuming that this is the case again leads to equations that are consistent with known results as to how human beings perceive colors.

**The Perceptual Information Content of a Photon**

Summarizing the discussion above, my hypothesis is that what is perceived as light is actually a visual representation of the number of bits required to encode the amount of energy observed. In the case of measuring the luminosity of a light source, this hypothesis works out perfectly, since it implies that perceived luminosity is given by the logarithm of actual physical luminosity (see below for the relevant analysis). In the case of an individual photon, it implies that the perceived luminosity of the photon should be given by .

However, we also know that at a certain point on either side of the visible spectrum of light, perception falls off, meaning that as the frequency of a source increases past blue, our response vanishes, making the light invisible. Similarly, as the frequency of a source decreases below red, the light will again eventually become invisible. This implies that we’ll need to adjust our equation to account for the fact that at both ends of the visible spectrum of light, the perceived luminosity should approach zero. We can accomplish this by adding a “weight” given by,

,

where is the frequency of the photon in question, is the minimum perceptible frequency (i.e., the minimum visible frequency of red) and is the maximum perceptible frequency (i.e., the maximum visible frequency of blue/violet). Combining these two equations (but dropping the constant), we have the following equation relating the frequency of a photon to its perceived luminosity:

.

Plugging in the reasonable values of THz and THz, we obtain the attached graph.

Note that this equation implies that perceived luminosity is maximized around a frequency of 607 THz, right at the boundary of the the green and blue portions of the visible spectrum, which is consistent with known results as to how human beings perceive differences in luminosities across the color spectrum.

In summation, whether or not we’re living in The Matrix, assuming that human beings perceive the world around us using encoded representations implies equations that are consistent with known results as to how we actually perceive the external world. Specifically, if we assume that what we perceive is information itself, and that the information content of a perceptual signal is given by the logarithm of the physical stimulus energy, then it seems that, as a general matter, we produce equations that are consistent with experimental data as to how human beings react to external stimuli.

I’ve published a working paper demonstrating the accuracy of my model of AI when applied to four UCI datasets, which you can find on my researchgate homepage.

Over the four classification problems, the categorization algorithm had an average success rate of 92.833%, where success is measured by the percentage of categories that are consistent with the hidden classification data. Over the four classification problems, the prediction algorithm had an average success rate of 93.497%, where success is measured by the percentage of predictions that are consistent with the hidden classification data. All of the code necessary to run these algorithms, and apply them to the training data, is available on my researchgate homepage.

I’ve also attached a pdf of the working paper here.

A New Model of Artificial Intelligence: Application to Data I

The full working paper for my new model of artificial intelligence is available on researchgate here. I’ve also attached a PDF.

Regards,

Charles

Below I’ll present a vectorized categorization algorithm, and a vectorized prediction algorithm, each of which have a worst-case, low-degree polynomial run time, even in extremely high dimensions. Together, these two algorithms allow for almost instantaneous inferences to be drawn from an enormous number of observations on an ordinary consumer device. As a result, I believe these algorithms will facilitate the commoditization of artificial intelligence, potentially sparking a revolution in the field, and a sea-change in the types of products available to consumers.

Specifically, I’ll show how these algorithms can be used to predict complex random-walks, predict projectile paths in three-dimensions, and classify three-dimensional objects, in each case making use of inferences drawn from millions of underlying data points, all using low-degree polynomial run time algorithms that can be executed on an ordinary consumer device. All of the code necessary to run these algorithms is attached.

**Vectorized Categorization and Prediction**

In the post below entitled, “Fast N-Dimensional Categorization Algorithm,” I presented a categorization algorithm that can categorize a dataset of N-dimensional vectors with a worst-case run time that is , where is the number of items in the dataset, and is the dimension of each vector in the dataset. The categorization algorithm I’ll present in this post is identical to the algorithm I presented below, except this algorithm skips all of the steps that cannot be vectorized. As a result, the vectorized categorization algorithm has a worst-case run time that is , where is the number of items in the dataset. The corresponding prediction algorithm is basically unchanged, and has a run time that is worst-case . This means that even if our underlying dataset contains millions of observations, if it is nonetheless possible to structure these observations as high-dimensional vectors, then we can use these algorithms to produce nearly instantaneous inferences, despite the volume of the underlying data.

As a practical matter, on an ordinary consumer device, the dimension of the dataset will start to impact performance for , and as a result, the run time isn’t truly independent of the dimension of the dataset, but will instead depend upon on how the vectorized processes are implemented by the language and machine in question. Nonetheless, the bottom line is that these algorithms allow ordinary, inexpensive consumer devices to almost instantaneously draw complex inferences from millions of observations, giving ordinary consumer devices access to the building blocks of true artificial intelligence.

**Predicting Random Paths**

Let’s begin by predicting random-walk style data. This type of data can be used to represent the states of a complex system over time, such as an asset price, a weather system, a sports game, the click-path of a consumer, or the state-space of a complex problem generally. As we work through this example, I’ll provide an overview of how the algorithms work, in addition to discussing their performance and accuracy. Note that if you’d like to test any of the data discussed in this article, you can use the scripts I’ve attached, which contain the same code used to generate the training data presented in the examples.

The underlying dataset will in this case consist of two categories of random paths, though the categorization algorithm and prediction algorithm will both be blind to these categories. Specifically, our training data consists of 1,000 paths, each of which contains 10,000 points, for a total of 10,000,000 observations. Of those 1,000 paths, 500 will trend upward, and 500 will trend downward. The index of the vector represents time, and the actual entry in the vector at a given index represents the value of the path at that time. We’ll generate the upward trending paths by making the value slightly more likely to move up than down as a function of time, and we’ll generate the downward trending paths by making the value slightly more likely to move down than up as a function of time.

For context, Figure 1 shows both a downward trending path and an upward trending path, each taken from the dataset.

Figure 2 shows the full dataset of 1,000 paths, each of which begins at the origin, and then traverses a random path over 10,000 steps, either moving up or down at each step. The upward trending paths move up with a probability of .52, and the downward trending paths move up with a probability of .48.

After generating the dataset, the next step is to construct two data trees that we’ll use to generate inferences. One data tree is comprised of elements from the original dataset, called the **anchor tree**, and the other data tree is comprised of threshold values, called the **delta tree**.

The anchor tree is generated by repeatedly applying the categorization algorithm to the dataset. This will produce a series of subcategories at each depth of application. The top of the anchor tree contains a single nominal vector, and is used by the algorithm for housekeeping. The categories generated by the first application of the categorization algorithm have a depth of 2 in the anchor tree; the subcategories generated by two applications of the categorization algorithm have a depth of 3 in the anchor tree, and so on. The algorithm selects an **anchor vector **as representative of each subcategory generated by this process. As a result, each anchor vector at a depth of 2 in the anchor tree represents a category generated by a single application of the categorization algorithm, and so on.

As a simple example, assume our dataset consists of the integers {1,2,8,10}. In this case, the first application of the categorization algorithm generates the categories {1,2} {8} {10}, with the anchors being 1, 8, and 10. This is the actual output of the categorization algorithm when given this dataset, and it turns out that, in this case, the algorithm does not think that it’s appropriate to generate any deeper subcategories. For a more detailed explanation of how this whole process works, see the post below entitled, “Predicting and Imitating Complex Data Using Information Theory.”

As a result, in this case, the anchor tree will contain the nominal vector at its top, and then at a depth of 2, the anchors 1,8, and 10 will be positioned in separate entries, indexed by different widths.

Represented visually, the anchor tree is as follows:

(Inf)

(1) (8) (10)

Each application of the categorization algorithm also produces a value called **delta**, which is the maximum difference tolerated for inclusion in a given category. In the example given above, delta is 1.0538. Specifically, if is in a category represented by the anchor vector , then it must be the case that , where is the sufficient difference associated with the category represented by . That is, is the difference between two vectors which, when met or exceeded, we treat as sufficient cause to categorize the vectors separately. The value of delta is determined by making use of processes I’ve developed that are rooted in information theory, but in short, delta is the natural, in-context difference between elements of the category in question. The value has the same depth and width position in the delta tree that the associated anchor vector has in the anchor tree. So in our example above, 1 and 2 are in the same category, since , but 8 and 10 are not, since .

Since the categorization algorithm is applied exactly once in this case, only one value of delta is generated, resulting in the following delta tree:

(Inf)

(1.0538) (1.0538) (1.0538)

Together, these two trees operate as a compressed representation of the subcategories generated by the repeated application of the categorization algorithm, since we can take a vector from the original dataset, and quickly determine which subcategory it could belong to by calculating the difference between that vector and each anchor, and testing whether it’s less than the applicable delta. This allows us to approximate operations on the entire dataset using only a fraction of the elements from the original dataset. Further, we can, for this same reason, test a new data item that is not part of the original dataset against each anchor to determine to which subcategory the new data item fits best. We can also test whether the new data item belongs in the dataset in the first instance, since if it is not within the applicable delta of any anchor, then the new data item is not a good fit for the dataset, and moreover, could not have been part of the original dataset. Finally, given a vector that contains out of values, we can then predict the missing values by substituting those missing values using the corresponding values in the anchors in the anchor tree, and then determining which substitution minimized the difference between the resulting vector and the anchors in the tree.

For example, if , and , then we would substitute the two missing values in the input vector , using the last two dimensions of an anchor vector , producing the prediction vector . We do this for every anchor in the anchor tree, and test whether . If the norm of the difference between the anchor vector and the prediction vector is less than the applicable delta, then the algorithm treats that prediction as a good prediction, since the resulting prediction vector would have qualified for inclusion in the original dataset. As a result, all of the predictions generated by the algorithm will contain all of the information from the input vector, with any missing information that is to be predicted taken from the anchor tree.

This implies that our input vector could be within the applicable delta of multiple anchor vectors, thereby constituting a match for each related subcategory. As a result, the prediction algorithm returns both a single best-fit prediction, and an array of possible-fit predictions. That is, if our input vector produces prediction vectors that are within the applicable delta of multiple anchors, then each such prediction vector is returned in the array of possible fits. There will be, however, one fit for which the difference between the input vector and the applicable prediction vector is minimized, and this is returned by the algorithm as the best-fit prediction vector.

Returning to our example, each path is treated as a 10,000 dimensional vector, with each point in a path represented by a number in the vector. So in this case, the first step of the process will be to generate the data trees populated by repeated application of the categorization algorithm to the dataset of path vectors. Because the categorization algorithm is completely vectorized, this process can be accomplished on an ordinary consumer device, despite the high dimension of the dataset, and the relatively large number of items in the dataset. Once this initial step is completed, we can begin to make predictions using the data trees.

We’ll begin by generating a new, upward trending path, that we’ll use as our input vector, that again consists of observations. Then, we’ll incrementally increase , the number of points from that path that are given to the prediction algorithm. This will allow us to measure the difference between the actual path, and the predicted path as a function of , the number of observations given to the prediction algorithm. If , giving the prediction algorithm a completely empty path vector, then the prediction algorithm will have no information from which it can draw inferences. As a result, every path in the anchor tree will be a possible path. That is, with no information at all, all of the paths in the anchor tree are possible.

In this case, the categorization algorithm compressed the 1,000 paths from the dataset into 282 paths that together comprise the anchor tree. The anchor tree has a depth of 2, meaning that the algorithm did not think that subdividing the top layer categories generated by a single application of the categorization algorithm was appropriate in this case. This implies that the algorithm didn’t find any local clustering beneath the macroscopic clustering identified by the first application of the categorization algorithm. This in turn implies that the data is highly randomized, since each top layer category is roughly uniformly diffuse, without any significant in-category clustering.

This example demonstrates the philosophy underlying these algorithms, which is to first compress the underlying dataset using information theory, and then analyze the compressed dataset efficiently using vectorized operations. In this case, since the trees have a depth of 2 and a width of 282, each prediction requires at most vectorized operations, despite the fact that the inferences are ultimately derived from the information contained in observations.

Beginning with observations, each path in the anchor tree constitutes a trivial match, which again can be seen in Figure 2. That is, since the algorithm has no observations with which it can generate an inference, all of the paths in the anchor tree are possible. Expressed in terms of information, when the input information is minimized, the output uncertainty is maximized. As a result, the prediction algorithm is consistent with common sense, since it gradually eliminates possible outcomes as more information becomes available.

Note that we can view the predictions generated in this case as both raw numerical predictions, and more abstract classification predictions, since each path will be either an upward trending path, or a downward trending path. I’ll focus exclusively on classification predictions in another example below, where we’ll use the same categorization and prediction algorithms to predict which class of three-dimensional shapes a given input shape belongs to.

Returning to the example at hand, I’ve set , and Figure 3 shows the actual path of the new input vector, the average path generated by taking the simple average of all of the predicted paths, and the best-fit path. In this case, the prediction algorithm has only .5% of the path information, and not surprisingly, the average of all predicted paths is flat, since the array of predicted paths contains paths that trend in both directions. In fact, in this case, the array of predicted paths contains the full set of 282 possible paths, implying that the first 50 points in the input path did not provide much information from which inferences can be drawn, though the best-fit path in this case does point in the right direction.

For , the actual path, average path, and best-fit path, all clearly trend in the same, correct direction, as you can see in Figure 4. Figure 5 contains the full array of predicted paths for , which consists of 173 possible paths. Though the average path, and best-fit path both point in the correct, upward trending direction, the set of possible paths clearly contains a substantial number of downward trending paths.

Figure 6 shows the average error between the best-fit path and the actual path as a percentage of the value in the actual path, as a function of . In this case, Figure 6 reflects only the portion of the data actually predicted by the algorithm, and ignores the first points in the predicted path, since the first points are taken from the input vector itself. That is, Figure 6 reflects only the values taken from the anchor tree, and not the first values taken from the input vector itself. Also note that the error percentages can and do in this case exceed 100% for low values of .

As expected, the best-fit path converges to the actual path as increases, implying that the quality of prediction increases as a function of the number of observations. Figure 7 shows the same average error calculation for 25 randomly generated input paths.

Despite the overall convergence of the predictions to the actual path, and the ability of the algorithm to quickly classify an input path as trending upward or downward, this type of data is by its nature incapable of exact prediction. As a result, there is an inherent limit to the precision of any predictions made using this type of data, and the prediction algorithm I’ve presented is no exception, and is, therefore, necessarily going to generate some appreciable amount of error given data that is this randomized. In contrast, in the next section, we’ll predict three-dimensional projectile paths, which allow for tighter predictions.

As noted above, the prediction algorithm is also capable of determining whether a given input vector is a good fit for the underlying data in the first instance. In this case, this means that the algorithm can determine whether the path represented by an input vector belongs to one of the categories in the underlying dataset, or whether the input vector represents some other type of path that doesn’t belong to the original dataset. For example, if we generate a new input vector produced by a formula that has a .55 probability of decreasing at any given moment in time, then for , the algorithm returns the nominal vector at the top of the anchor tree as its predicted vector, indicating that the algorithm thinks that the input vector does not belong to any of the underlying categories in the anchor tree.

This ability to not only predict outcomes, but also identify inputs as outside of the scope of the underlying dataset helps prevent bad predictions, since if an input vector is outside of the scope of the underlying dataset, then the algorithm won’t make a prediction at all, but will instead flag the input as a poor fit. This also allows us to expand the underlying dataset by providing random inputs to the prediction function, and then take the inputs that survive this process as additional elements of the underlying dataset. For more on this approach, see the post below entitled, “Predicting and Imitating Complex Data Using Information Theory.”

**Categorizations and Predictions in Higher Dimensional Spaces**

*Predicting Projectile Paths*

The same algorithms that we used in the previous section to predict paths in a two-dimensional space can be applied to an N-dimensional space, allowing us to analyze high-dimensional data in high-dimensional spaces. We’ll begin by predicting somewhat randomized, but nonetheless Newtonian projectile paths in three-dimensional space.

In this section, our training data consists of 750 projectile paths, each containing 3000 points in Euclidean three-space, resulting in vectors with a dimension of . The equations that generated the paths in the training data are Newtonian equations of motion, with bounded, but randomly generated x and y velocities, no z velocity other than downward gravitational acceleration, fixed initial x and y positions, and randomly generated, but tightly bounded initial z positions. Figure 8 shows a few of the paths from the dataset.

Note that in this case, we’re predicting a path that has a simple closed form formula without using any interpolation. Instead, the algorithm uses the exact same approach outlined above, which compresses the dataset into a series of anchor vectors, which in this case represent Newtonian paths, and then takes a new curve and attempts to place it in the resulting anchor tree of vectors.

Just as we did above, first we’ll generate a new curve, which will be represented as a 9,000 dimensional vector that consists of 3,000 points in Euclidean three-space. Then, we’ll incrementally provide points from the new curve to the prediction algorithm, and measure the accuracy of the resultant predictions. Figure 9 shows the average path, best-fit path, and actual path for .

In this case, the percentage-wise average error between the predicted path and the actual path, shown in Figure 10, is much smaller than it was for the random path data. This is not surprising, given that each curve has a simple shape that is entirely determined by its initial conditions. As a result, so long as there is another curve in the anchor tree that has reasonably similar initial conditions, we’ll be able to accurately approximate, and therefore, predict the shape of the entire input curve.

*Shape and Object Classification*

We can use the same techniques to categorize three-dimensional objects. We can then take the point data for a new object and determine to which category it best belongs by using the prediction algorithm. This allows us to take what is generally considered to be a hard problem, i.e., three-dimensional object classification, and reduce it to a problem that can be solved in a fraction of a second on an ordinary consumer device.

In this case, the training data consists of the point data for 600 objects, each shaped like a vase, with randomly generated widths that come in three classes of sizes: narrow, medium, and wide. That is, each class of objects has a bounded width, within which objects are randomly generated. There are 200 objects in each class, for a total of 600 objects in the dataset. Each object contains 483 points, producing a vector that represents the object with a total dimension of . I’ve attached a representative image from each class of object as Figures 11, 12, and 13. Note that the categorization and prediction algorithm are both blind to the classification labels in the data, which are hidden in the entry of each shape vector.

As we did above, we’ll generate a new object, provide only part of the point data for that object to the prediction algorithm, and then test whether the prediction algorithm assigns the new object to a category with an anchor from the same class as the input object. That is, if the class of the anchor of the category to which the new input object is assigned matches the class of the input object, then we treat that prediction as a success.

I’ve randomly generated 300 objects from each class as a new input to the prediction algorithm, for a total of 900 input objects. I then provided the first $M = 1200$ values from each input vector to the prediction algorithm, producing 900 predictions. There are three possibilities for each prediction: success (i.e., a correct classification); rejection (i.e., the shape doesn’t fit into the dataset); and failure (i.e., an incorrect classification).

The results for each of the three classes is as follows:

**Class One:**

*Success Percentage: 300 / 300 = 100%. *

*Rejected Data Percentage: 0 / 300 = 0%.*

*Fail Percentage: 0 / 300 = 0%.*

**Class Two:**

*Success Percentage: 300 / 300 = 100%.*

*Rejected Data Percentage: 0 / 300 = 0%.*

*Fail Percentage: 0 / 300 = 0%.*

**Class Three:**

*Success Percentage: 300 / 300 = 100%.*

*Rejected Data Percentage: 0 / 300 = 0%.*

*Fail Percentage: 0 / 300 = 0%.*

These results are of course the product of not only randomly generated training data, but also randomly generated inputs. Different datasets, and different inputs could therefore produce different results. Nonetheless, the obvious takeaway is that the prediction algorithm is extremely accurate.

**The Future of Computation**

The Church-Turing Thesis is a hypothesis that asserts that all models of computation can be simulated by a Universal Turing Machine. In effect, the hypothesis asserts that any proposed model of computation is either inferior to, or equivalent to, a UTM. The Church-Turing Thesis is a hypothesis, and not a mathematical theorem, but it has turned out to be true for every known model of computation, though recent developments in quantum computing are likely to present a new wave of tests for this celebrated hypothesis.

In a post below entitled, “A Simple Model of Non-Turing Equivalent Computation,” I presented the mathematical outlines of a model of computation that is on its face not equivalent to a UTM. That said, I have not yet built any machines that implement this model of computation, so, not surprisingly, the Church-Turing Thesis still stands.

The simple insight underlying my model is that a UTM cannot generate computational complexity, and this can be proven mathematically (see the link above for a simple proof). This means that the computational complexity (i.e., the Kolmogorov Complexity) of the output of a UTM is always less than or equal to the complexity of the input that generated the output in question. This simple lemma has some alarming consequences, and in particular, it suggests that the behaviors of some human beings might be the product of a non-computable process, perhaps explaining how it is that some people are capable of producing ideas, predictions, and works of art, that appear to come from out of nowhere.

Since we can, as a theoretical matter, map a given input string to an output string that has a higher computational complexity than the input string, any device that actually does this as a physical matter over an infinite set of outputs is by definition not equivalent to a UTM. Expressed symbolically, , and , for some infinite set of outputs , where is the Kolmogorov complexity. If the set of possible outputs is infinite, then the device can consistently generate complexity, which is not possible using a UTM. If the device generates only a finite amount of complexity, then the device can of course be implemented by a UTM that has some complexity stored in the memory of the device. In fact, this is exactly the model of computation that I’ve outlined above, where a compressed, specialized tree is stored in the memory of a device, and then called upon to supplement the inputs to the device, generating the outputs of the device.

The model of AI that I’ve presented above suggests that we can create simple, presumably cheap devices, that have specialized memories of the type I described above, that can provide fast, intelligent answers to complex questions, but nonetheless also perform basic computations. This would allow for the true commoditization of artificial intelligence, since these devices would presumably be both small and inexpensive to manufacture. This could lead to a new wave of economic and social changes, where cheap devices, possibly as small as transistors, have the power to analyze a large number of complex observations and make inferences from them in real-time.

In the post below entitled, “Predicting and Imitating Complex Data Using Information Theory,” I showed how these same algorithms can be used to approximate both simple, closed-form functions, and complex, discontinuous functions that have a significant amount of statistical randomness. As a result, we could store trees that implement everyday functions found in calculators, such as trigonometric functions, together with trees that implement complex tasks in AI, such as image analysis, all in a single device that makes use of the algorithms that I outlined above. This would create a single device capable of both general purpose computing, and sophisticated tasks in artificial intelligence. We could even imagine these algorithms being hardwired as “smart transistors” that have a cache of stored trees that are then loaded in real-time to perform the particular task at hand.

All of the algorithms I presented above are obviously capable of being simulated by a UTM, but as a general matter, this type of non-symbolic, stimulus and response computation can in theory produce input-output pairs that always generate complexity, and therefore, cannot be reproduced by a UTM. Specifically, if the device in question maps an infinite number of input strings to an infinite number of output strings that each have a higher complexity than the corresponding input string, then the device is simply not equivalent to a UTM, and its behavior cannot be simulated by a UTM with a finite memory, since a UTM cannot, as a general matter, generate complexity.

This might sound like a purely theoretical model, and it might be, but there are physical systems whose states change in complex ways as a result of simple changes to environmental variables. As a result, if we are able to find a physical system whose states are consistently more complex than some simple environmental variable that we can control, then we can use that environmental variable as an input to the system. This in turn implies that we can consistently map a low-complexity input to a high-complexity output. If the set of possible outputs appears to be infinite, then we would have a system that could be used to calculate functions that cannot be calculated by a UTM. That is, any such system would form the physical basis of a model of computation that cannot be simulated by a UTM, and a device capable of calculating non-computable functions.

*Octave Scripts*:

*Data Sets*:

In this post, I’ll present an unsupervised algorithm that can, in polynomial time, extract three-dimensional features from a two-dimensional image. The results are quite good, especially given the run time. This algorithm could have applications in computer vision, in particular medical imaging, and perhaps other fields, but even as a novelty, it’s a fun algorithm that can be run on an ordinary consumer device.

**Feature Extraction**

Several months ago, I presented an algorithm that can quickly identify features in an image with no prior information, which I explained in some detail in the post below this one entitled, “Fast Unsupervised Image Categorization Algorithm”. In short, the feature identification algorithm works by partitioning an image into maximally distinct regions by making use of methods rooted in information theory.

By changing the equation that governs the stopping condition in the feature identification algorithm, we can allow the algorithm to run longer, and identify fine-grain features in an image, similar to an edge detection algorithm. The fine-grain feature identification algorithm also has a polynomial run-time, but it is much slower, and as a result, I generally do not use it, though it was the original version of the algorithm.

However, I realized that the fine-grained algorithm is useful for making detailed distinctions between foreground and background. This is because the feature identification algorithm assigns a probability to each feature that gives the likelihood that the feature in question is part of the background of an image. It turns out that this probability is reliable, and so the background of an image generally has a probability of approximately 0, and the foreground generally has a probability of approximately 1. By turning this probability into a depth parameter, we can map a two-dimensional image into a three-dimensional space, where a background feature has a depth of 1, and a foreground feature has a depth of 0 (i.e., one minus the foreground probability), in turn producing a three-dimensional rendering of the two-dimensional image, where the background features are recessed at a greater depth than the foreground features. If we use the fine-grain feature identification algorithm, the features will be smaller in scale, allowing for tighter depth information.

The examples I’ve posted were generated using very low-resolution photos, of around around 250 pixels x 250 pixels. Higher resolution photos produce very fine features, and correspondingly rich depth information, but of course take much longer to render.

This is of course not the algorithm of choice if you’re looking for a mathematically exact rendering. But, it could be useful whenever a fast approximation of depth is required, and the only tool available is information generated by an ordinary two-dimensional camera. In any case, it is certainly a useful heuristic for depth given two-dimensional information, and at a minimum, it is a fun algorithm to experiment with that produces visually meaningful results.

I’ve attached four scripts that you’ll need to run the algorithm, and the rest of the scripts you’ll need are available in the post just below this one.

In this post, I’m going to show how we can combine the image feature recognition algorithm I presented a few months ago, with the categorization and prediction algorithms I presented in the two posts below, to produce an algorithm that can quickly categorize images, without any supervision. The final algorithm I’ll present is O(n), where n is the number of pixels in the image we’re attempting to categorize.

This algorithm works so well that it can distinguish between paintings by different artists, in just a few seconds. It can also distinguish between articles of clothing, and other objects, but the focus of this post will be on its ability to distinguish between artists, because, in my opinion, this shows that the algorithm is capable of unsupervised abstraction, allowing it to say without any guidance, what it is that distinguishes a Van Gogh from a Caravaggio.

These algorithms use the same underlying categorization and prediction algorithms that I used below to imitate complex three-dimensional shapes, predict data, and approximate complex functions. In this case, I’ve added some additional scripts that allow the underlying algorithms to analyze images. As a result, the underlying categorization and prediction algorithms are shaping up to be a set of generalized AI algorithms that appear to only require a “hook” into the subject matter in question.

Finally, note that all of the algorithms I’ll present in this post are worst-case polynomial in time as a function of the number of pixels in the image. That is, even the algorithms we’ll use to construct categories in the first instance are polynomial in time as a function of the number of pixels / images in the database. As a result, these algorithms allow for what I believe to be industrial quality image recognition to be implemented on a consumer device, and can probably allow industrial machines to conduct real-time analysis of high-resolution video.

**Summary of the Underlying Image Feature Identification Algorithm**

In the post below entitled, “Image Recognition with no Prior Information”, I presented an algorithm that can quickly identify features within an image, without any prior information about the image, or the categories to which the image might belong. At a high level, this algorithm takes an image, and partitions it into objectively distinct features. But because the algorithm operates with no prior information at all, it isn’t looking for a particular object, but is instead looking for every objectively distinct feature within an image. In short, this first algorithm attempts to identify all of the features within an image, but because this algorithm has no prior information, it can’t say what these features are, but can nonetheless identify them as distinct from one another. As a result, this algorithm is only the first step of the image processing algorithms I’ll present in this post.

Like all of the other work that is part of this project, the feature recognition algorithm is rooted in information theory, and operates based upon the assumption that objectively distinct features within an image will contain different amounts of information. Specifically, the feature identification algorithm subdivides an image in a way that maximizes the difference in the amounts of information contained in each subdivision of the image. I call each subdivision of an image generated by the algorithm a region. Though it might seem pedantic to distinguish between a region and a feature, I’d like to reserve the term feature to denote an actual feature of an image, as identified by a human, whereas a region is a portion of an image identified by the algorithm that should contain a feature.

The first, second, and third attached images are examples of an image that has been processed by the algorithm. The first image is a photo I took in Lake Como, Italy, and the second image shows the original image after being processed, with the brightness of each region adjusted by the algorithm to reflect how likely it thinks the region is to be part of the foreground of the image. The brighter a given region is, the more likely the algorithm thinks that the region is part of the foreground of the image. You can get a sense of how the algorithm partitions an image by looking at the boundaries of the regions in the second image. The third image shows a specific region identified by the algorithm, which is in this case an individual standing on a diving board, which can be found around the center of the original image.

You can do this yourself using the following code:

pkg load communications

pkg load image

I = imread(“[IMAGE FILE]”);

[region_matrix N S_mat prob_mat] = identify_features(I);

imshow(ent_contrast_darken_file(I,N,region_matrix, prob_mat));

figure, imshow(extract_region(I,N,region_matrix,27));

**Constructing a Tractable Color Palette**

At a high level, the technique we’ll employ is straightforward: first we’re going to partition an image using the feature identification algorithm described above, and that’s going to produce a series of regions, which should correspond to features within the image. Then, we’re going to calculate the average color of each region, which will create a dataset of average colors. Then, we’re going to categorize those colors using the vector categorization algorithm I presented in the two posts just below this one.

In this section, I’ll walk through applying this process to a single image. In the next section, we’ll apply it to a series of images. In the last section, I’ll present a vectorized implementation of this entire process that will allow us to categorize new images using an algorithm that is O(n).

I’ve attached a photograph of one of Van Gogh’s iconic self-portraits that we’ll use as our first subject.

First, we’ll count the number of unique colors in the image using a vectorized script I found on Matlab’s website, written by Steve Eddins, which is available here:

https://blogs.mathworks.com/steve/2008/01/31/counting-occurrences-of-image-colors/

I = imread(“[IMAGE FILE]”);

rgb_columns = reshape(I, [], 3);

[unique_colors, m, n] = unique(rgb_columns, ‘rows’);

color_counts = accumarray(n, 1);

size(unique_colors, 1)

The image contains 184064 unique colors, and since the image contains 768*608 = 466944 pixels, about 40 percent of the pixels contain unique colors. As a result, the image contains a very rich color palette, and therefore, quite a lot of color information for its size. Nonetheless, a human observer probably wouldn’t have much trouble quickly describing the color palette of the painting, which contains a lot of yellow, due to his large hat, with a pinkish and orange background, and some pale blues in his blazer.

This type of description is a form of compression, since we’ve taken a large amount of color information, and compressed that information into a tractable statement that is nonetheless a meaningful description of its subject. Similarly, the goal of this section will be to produce a compressed color palate that straddles the line the between a set of colors that is digestible by a human, and a set of colors that is useful to a machine. The color sets we’ll produce will be too large to be described in one sentence, but small enough to be expressed visually in a single color bar that can be quickly gleaned by a human. The intended end result is to have the categorization algorithm operate like an enhanced human, that quickly compresses information, but nonetheless has access to details that a human observer would find overwhelming.

Continuing with the example of the self-portrait, first we’ll run the feature identification algorithm, which will produce a set of matrices we can use to partition the image into regions. Then, we’ll run a script that calculates and stores the average color of each region, which will produce a dataset of color vectors. Finally, we’ll run the categorization algorithm on that dataset of color vectors, which will produce a core set of color vectors that can be thought of as forming a basis for the colors in the image. I call these basis vectors the anchors of their color categories, since the categorization algorithm treats each anchor vector as the geometric center of a category, meaning that all other vectors in the category defined by the anchor are within some distance, delta, that the algorithm calculates based upon a variety of factors that I explain in the two posts below this one.

All of this can be done using the following code:

[region_matrix N S_mat prob_mat] = identify_features(I);

figure, imshow(ent_contrast_darken_file(I,N,region_matrix,prob_mat));

data_array = calculate_average_feature_color(I,N,region_matrix, prob_mat, 0, 0);

[data_categories_array category_vec anchor_array H_final delta] = optimize_categories_3D(data_array,0);

color_bar_compressed = display_color_array(anchor_array, data_categories_array);

figure, imshow(color_bar_compressed)

The feature identification algorithm generates 94 regions, and the color bar shows the compressed set of colors generated by the categorization algorithm, which consists of 44 anchor colors. Visually, this set of colors is clearly representative of the underlying painting, despite containing only about .02% of the unique colors in the image. As a result, even though average color is generally a very lossy metric for an image, because we’ve taken the average color of what are objectively distinct features of the image, we’ve ended up with meaningful color information, while nonetheless achieving an enormous amount of compression. Not to boast, but this is actually quite remarkable, since the algorithm has taken over one hundred thousand colors from an image, and produced, without any supervision, a set of 44 colors that are perfectly representative of the underlying image. Moreover, it would almost certainly be impossible for a human to sift through all of the colors in the original image and select a comparable set of colors. At the same time, these are the colors that a human probably would select if they could. As a result, this process generates the same type of compression a human-generated, verbal summary achieves, but on a mechanized scale, and expressed nonverbally, in the form of an image. Interestingly, it also suggests that nonverbal summaries of this type could allow machines to convey complex information to their human counterparts.

This of course does not preclude more sophisticated measures from being used in lieu of average color. Instead, I’m presenting this as a simple example of a more generalized approach to image recognition that consists of three primary steps:

(1) identify the features of an image using the feature identification algorithm described above;

(2) analyze the properties of each resulting region, producing a metric;

(3) categorize the resulting metrics.

If our metric is visually meaningful, then the categories we’ll construct in step three will cause the metrics associated with similar looking features to be categorized together, thereby creating categories of similar looking features. It turns out that average color works quite well, though there are undoubtedly variations on this theme that might be more useful in other contexts.

**Categorizing Images**

By applying the process above to a category of similar images, we can construct a palette that is representative of an entire category of images. In this case, we’re going to construct two palettes: one generated by analyzing a series of paintings by Van Gogh, and another generated by analyzing a series of paintings by John Singer Sargent. These two painters worked with somewhat similar color schemes, and to a non-expert like myself, a superficially similar impressionistic style. Nonetheless, the paintings I’ve selected are distinguishable, and as a result, the two categories of paintings will produce different color palettes. We’ll analyze a set of 8 paintings by each artist, for a total of 16 images, each of which are attached as the last 16 images.

We can produce the palettes using the following code:

directory_path_A = “[Van Gogh]”;

directory_path_D = “[Sargent]”;

num_images_A = 8;

num_images_D = 8;

[category_tree1 delta_tree1 anchor_tree1 data_array1] = generate_single_path_img_data(directory_path_A, num_images_A);

[category_tree4 delta_tree4 anchor_tree4 data_array4] = generate_single_path_img_data(directory_path_D, num_images_D);

[data_categories_array1 category_vec1 anchor_array1 H_final1 delta1] = optimize_categories_3D(data_array1,0);

[data_categories_array4 category_vec4 anchor_array4 H_final4 delta4] = optimize_categories_3D(data_array4,0);

color_bar_compressed1 = display_color_array(anchor_array1, data_categories_array1);

color_bar_compressed4 = display_color_array(anchor_array4, data_categories_array4);

figure, imshow(color_bar_compressed1)

figure, imshow(color_bar_compressed4)

The Van Gogh set of images generates 711 features, ultimately compressed into a set of 268 anchor colors, while the Sargent set of images generates 600 features, which were compressed into a set of 243 anchor colors. Yet again, these algorithms achieve an incredible amount of compression, taking millions of colors in the underlying images, and generating a comparatively minuscule set of colors that a human being can easily observe and intuitively understand, albeit in a nonverbal manner.

The Van Gogh palette consists of an array of rich, and bright pastels, whereas the Sargent palette consists of what I thought to be a surprisingly dark set of colors, mixed in with brighter colors that are nonetheless closer to the grey portion of the spectrum when compared to the vibrant colors that make up the Van Gogh palette.

If we were to take yet another Van Gogh painting, and compare the color palette of that new painting to the Van Gogh palette we just produced, it would probably be a decent match. Moreover, the color palette of the new Van Gogh would almost certainly be a better match for the existing Van Gogh palette than the Sargent palette. As a result, if we’re given a new image, and we assume that the new image is either a Van Gogh or a Sargent, then by comparing the color palette of the new image to the two color palettes we just produced, we should be able to categorize that new image as either a Van Gogh or a Sargent.

The image categorization algorithm I’ll present in this section is O(n*log(n)), since we’ll need to run the feature identification algorithm on the new input image in order construct its color palette. In the next section, I’ll present a vectorized version of this algorithm that is O(n), that takes into account the color of every pixel in the image. As a result, the vectorized algorithm is not only faster, but it’s also more powerful, since it takes into account all of the color information within an image. Nonetheless, the intuition and inspiration for the vectorized algorithm comes from this longer form process.

Let’s begin by generating a color palette for the beardless Van Gogh self-portrait, which I’ve attached, using the same code above from the previous section. Though this image was not part of the database that generated the Van Gogh palette, just looking at it, we can already see that it’s probably going to fit well. Not surprisingly, the resultant color palette looks a lot like a subset of the color palette for the entire Van Gogh series. As a result, it makes sense to assume that this set of colors will be a better fit for the Van Gogh palette than for the Sargent palette.

We can test this hypothesis using a prediction function I presented in the previous post, that takes a new data item, and attempts to categorize it into an existing dataset that’s already been analyzed using the categorization algorithm. At a high level, what the prediction algorithm does is attempt to place the new data item into an expanded set of anchor vectors that are structured as a tree. If it can place the new data item somewhere in the tree, then the new data item is a good match for the dataset. If it fails to place the new data item in the tree, then the new data item is not a good match for the dataset. For a full explanation as to how and why the prediction algorithm works, see the two posts just below this one.

In this particular case, we can think of the prediction algorithm as attempting to fit each new feature color into an existing dataset of anchor colors. The greater the number of feature colors that match with a given dataset of anchor colors, the more confident we can be that the new image is, as a whole, a match for that dataset in terms of its color palette. Of course, knowing that two images have similar color palettes is not, as a general manner, enough information to conclude that the two images belong to the same category of images or objects. But in this case, we’re tacitly assuming that the painting is either a Van Gogh or a Sargent, and then determining to which category the painting belongs by comparing a compressed representation of its color palette to the Van Gogh and Sargent palettes we just compiled.

This can be done using the following code:

[region_matrix N S_mat prob_mat img_data_array score_vector1 delta_vector1] = AB_test_image(I, category_tree1, anchor_tree1, delta_tree1);

[region_matrix4 N4 S_mat4 prob_mat4 img_data_array4 score_vector4 delta_vector4] = AB_test_image(I, category_tree4, anchor_tree4, delta_tree4);

sum(score_vector1 != Inf)

sum(score_vector4 != Inf)

In short, the underlying prediction algorithm will attempt to place the average color of each region from the new image into each dataset. If it succeeds, it will return a score of 0, and if it fails, it will return a score of Inf, which indicates that the algorithm thinks that the new color doesn’t belong to the dataset. The last two lines of code count how many regions matched with the two datasets. In this case, the new Van Gogh matched with the Van Gogh dataset 50 times, and matched with the Sargent dataset 40 times. Each test involved attempting to place 80 regions, meaning that in both cases, a significant number of regions failed to categorize.

Though this approach works reasonably well, note that it tests whether the average color of an entire region is a good fit for the color palette in question. As a result, the test does not take into account the vast majority of color information in an image, since each region is going to contain a large number of pixels. In the next section, we’ll vectorize this algorithm, which will allow us to quickly test every pixel in an image, thereby creating a simple filter that can test whether the colors in an image fit well within the complex, and discontiguous sets of colors that comprise the palettes we’ve constructed. Ultimately, this will allow us to quickly categorize images using a simple filtering process.

**Vectorized Image Categorization**

The first three steps in the process above cannot, to my knowledge, be vectorized. That is, if we want to construct a color palette for a database of images, we need to run the feature identification algorithm, calculate the average color of each resulting feature, and then categorize the resulting dataset of colors. However, if we’d like to compare the colors within an image to that color palette, after it’s already been produced, then we can take advantage of the ready-baked vectorized procedures of Matlab and Octave, and possibly other languages as well.

In short, what we’ll do is simply test whether each pixel in an image is roughly within delta of at least one anchor color. If the pixel is within delta of some anchor, then it will pass the filter. If it is not within delta of some anchor, then it will not pass the filter. By calculating the percentage of pixels that survive filtration using this method, we can measure what percentage of pixels in the image are a match for the color palette in question. This in turn allows us to quickly determine to which category of images a given input image belongs, by choosing the category associated with the filter that had the highest pixel survival rate. That is, the input image is best suited in terms of color with the underlying category of images that generated the filter that does the least damage.

For example, if we’d like to determine whether a given image is a Van Gogh or a Sargent, we can use the color palettes we generated above, and test that new image using a Van Gogh filter and a Sargent filter. The filter with the higher pixel survival rate is the better fit, and therefore tells us which of the two painters is most likely to have held the brush.

Returning to the beardless Van Gogh portrait we tested above, we can apply a Van Gogh filter and a Sargent filter to that image using the following code:

%the Van Gogh filter

[not_mask mask] = generate_category_mask(I, anchor_tree1, delta_tree1);

I_Post = I.*mask;

figure, imshow(I_Post)

num_on_pixels = sum(mask(:) == 1);

[row col x] = size(I);

num_pixels = row*col;

survival_rate1 = num_on_pixels/num_pixels

%the Sargent filter

[not_mask mask] = generate_category_mask(I, anchor_tree4, delta_tree4);

I_Post = I.*mask;

figure, imshow(I_Post)

num_on_pixels = sum(mask(:) == 1);

[row col x] = size(I);

num_pixels = row*col;

survival_rate4 = num_on_pixels/num_pixels

The survival rate for the Van Gogh filter is 0.29383, or roughly 30% of the pixels, whereas the survival rate for the Sargent filter is 0.13676, or roughly 14% of the pixels.

As a general matter, this approach works astonishingly well, and gives wide margins of survival rates even in cases like this, where we’re distinguishing between similar images that are, frankly, difficult for even some humans to categorize. For example, if we’d like to A/B test for whether an image contains one of two dresses, then this approach will work, assuming that the two dresses have objectively distinct color palettes, even if they’re superficially quite similar to a human observer. Since this approach is not necessarily restricted to color, I believe this algorithm could mark the beginning of an entirely new approach to image recognition.

Octave Scripts:

calculate_average_feature_color