Relationships, Geometry, and Artificial Intelligence


By Gunnar Carlsson December 3, 2018

In their very intriguing paper, Peter Battaglia and his colleagues, posit that in order for expert system (AI) to accomplish the capabilities of human intelligence, it needs to be able to calculate with extremely structured information types, such as the ones people deal with. This means that AI needs to be able to deal with info about relationships in between objects, which are efficiently encoded as charts, i.e. collections of nodes and edges in between these nodes, and likewise associates connected to the nodes and edges.

A basic example would be images, where one considers graphs which are formed like a square grid, with the pixel worths being the characteristic connected to a particular node in the grid. The authors go on to explain the structure they have developed for calculating with graph structured information, in specific describing how neural networks for this type of calculation can be constructed.

Embedded in this work is the following concern:

Where do the graphs originate from?

The question recognizes the reality that the majority of information is “sensory” in character, i.e. it is obtained by measuring mathematical amounts and taping their worths as collaborates of vectors in a high dimensional space, and consequently do not come equipped with any explicit underlying chart structure.

It follows that in order to achieve human like intelligence in studying such information, it is needed to build chart or relational structures from sensory data, and also that performing such building and constructions is a critical active ingredient in human intelligence.

The objective of this post is to explain an approach for performing precisely this job, and to describe how neural networks can be built directly from the output of such techniques. We can summarize the technique as follows.

  1. A key ingredient is a geometry or metric on the set of features or sensors that are capturing the sensory data.
  2. The second key ingredient is that of a covering of the set of features by subsets. The coverings typically are chosen to be well related to the geometry in a suitable sense.
  3. The geometry and related coverings are used in conjunction to construct graph structures from the features defining the data.
  4. They are also used to construct analogues of the pooling constructions used in convolutional neural networks, which permit the learning to be based on various levels of resolution.  

The interested reader ought to consult the Topological Data Analysis Based Approaches to Deep Learning and Topological Approaches to Deep Learning where the buildings are talked about in detail.

We highly concur with the perspective revealed in the DeepMind/Google paper that their work turns down the false choice in between “hand-engineering” and “end-to end” approaches to finding out, and rather argues for the acknowledgment that the methods are complementary and need to be dealt with as such. What follows is that we should be developing a broad range of tools and methods that allow hand-engineering of neural networks. The approaches explained in the DeepMind/Google paper are an action in that instructions, as are papers on TDA-based Approaches to Deep Learning and Topological Approaches to Deep Learning.

We likewise observe that the benefits of such an approach exceed improving the precision and speed of neural net calculations. Once the point of view of the DeepMind/Google paper is adopted, one will have the ability to produce algorithms which realize the idea of machine-human interaction. In specific, they will be more transparent, so that one can understand their internal performance, and also supply more intricate “human-readable” outputs, with which people can interact. This paper on the Exposition and Interpretation of the Topology of Neural Networks makes up a step in that direction.

Finally, we observe that in order to fully recognize this vision, it is important not just to make the calculation plan human understandable, however also the input information. Without a better understanding of what is in the information, the improvements explained by Battaglia and his associates will just get us part of the method to totally actualizing expert system. Topological techniques supply tools for comprehending the relational and qualitative aspects of information sets, and ought to be utilized in the understanding of the algorithms which evaluate them.

Graphs from geometry

In the case of images, the graph structure on the set of characteristics (pixels) is that of a square grid, with some diagonal connections consisted of.

Our first observation is that the grid is actually included in a constant geometric things, particularly the plane. The importance of the plane is that it is equipped with a concept of range, i.e. a metric in the mathematical sense (see Reisel’s work here). The set of vertices are selected so as to be well distributed through a square in the plane, so that every point in the square is close to among the vertices. In addition, the connections in the graph can be presumed from the range. This is so because if we let r denote the distance from a vertex to its nearby next-door neighbor (this is independent of what vertex we choose because of the regularity of the layout), then 2 vertices are linked if their range is < < √ 2r.

So, the chart is determined by:

(a) a choice of subset of the aircraft and

(b) a notion of distance in the aircraft

The value of this observation is that we frequently have sets of attributes or features that are geared up with metrics. Some fascinating examples were studied in this paper, where a research study of the analytical habits of spots in natural images was carried out. Among the findings was that there is a crucial function that determines the “angular choice” of a spot in the direction of a particular angle θ. As we differ θ, we get a set of functions which is geometrically laid out on a circle, which can be discretized and become the graph on the left listed below. A more comprehensive analysis showed a two-parameter set of functions that are set out in the kind of a geometric things called a Klein bottle, revealed with associated graph structure on the right listed below.

Analogous geometries can be found for 3D imaging and video imaging. The operate in the Exposition paper has suggested larger spaces consisting of the Klein bottle which will use to larger scale functions and more abstract items.

Purely data driven graph building and constructions

The buildings in the previous area rely on a prior knowledge concerning the functions or a comprehensive data analytic study of particular picking up technologies. To have an approach that is more generally applicable, without complex information analysis, it is needed to have methods that can rapidly produce an useful chart structure on the set of functions. Such a technique exists, and is explained in both the TDA paper as well as the Topology paper.

Suppose that we have an information matrix, where the rows are the samples or observations, and the columns are functions or qualities. Then the columns are vectors, and there are various notions of distance one might equip them with, consisting of high-dimensional Euclidean range (perhaps applied to suggest focused and stabilized columns), connection, angle, or Hamming range if the values are Boolean. There are generally lots of options. By selecting a distance threshold, we can build a chart whose nodes are the private features. The graph will be sparser than if we choose a bigger threshold if the limit is small. At this point we have a method for building a chart from sensory data. Even more, the chart explains a resemblance relation between the features, in the sense that if 2 functions are connected by an edge, we understand that their range (related to as a dissimilarity measure) is less than some fixed threshold, and therefore that they resemble each other.

This building and construction certainly makes good sense and generalizes the construction that is provided for images. It produces charts which are large in the sense that they have the very same number of nodes as we have features. They may likewise be dense in the sense that some nodes may have a very big variety of connections, which is not perfect from the point of view of calculation. We will address this problem via a kind of compression of the graph structure.

To understand this building, we will talk about another aspect of the deep learning pipeline for images, specifically the pooling layers. We describe it as follows.

The “pooling” refers to pooling values of subsets of pixels, specifically smaller sized squares. The photo listed below describes the circumstance, where we have covered the pixel grid by 9 smaller subgrids, each one 2 by two.

The idea of pooling is to produce a brand-new chart, this time three by three, with one node for each of the subgrids. The interesting point is that the development of the new graph can be described in regards to the crossways of the subgrids only. Particularly, the nodes corresponding to two subgrids are linked by an edge if and only if the subgrids have at least one pixel in common. The worth of this observation is that this requirement generalizes to the general scenario of a geometry on the space of features, provided that we can produce a covering comparable to the covering of this square by subgrids. We describe the collection of subgrids as a covering of the set of pixels, where by a covering of a set X we mean a household of subsets U of X so that every element of X is consisted of in some member of the U covering.

Topological information analysis supplies an organized method of producing such coverings, using a technique described as Mapper, which is described in the TDA and Topology documents.

The output of this building will be a family of charts, ⌈ 1 …, ⌈ r with the following properties.

  1. Each vertex in each graph corresponds to a family of features.
  2. The collections corresponding to vertices are small or localized in the assigned metric on the set of features.
  3. Each graph is low-dimensional in the sense that the number of vertices that lie in a clique is bounded above.
  4. The size of the sets of features corresponding to vertices grow as we move from ⌈to r. In other words, the graphs become coarser and coarser representations of the set of features.

The building yields a systematic method of producing chart representations of the sensory information offered from the information matrix. It ends up that one can go straight from this representation to neural network architectures.

In fact, one could even think of incrementally producing coverings using Mapper on the activation-values of the layers following the input layer. Sometimes recomputing the coverings and therefore the graph structure would allow the network to adapt its graph structure during training.

Constructing neural networks

The circular constructions given above can offer some really easy geometric constructions of neural network architectures adapted to them. In the case of the circular networks described above, an image of such a building would look as follows.

Similar buildings are offered for the Klein bottle geometries and others.

The building and construction of neural networks from the simply information driven buildings above is likewise rather basic. Since each node of each chart corresponds to a set of features from the data matrix, we can declare that a node ν in ⌈ r is linked by a directed edge to a node w in ⌈ r +1 if and just if the collection corresponding to ν and the collection corresponding to w contain at least one feature in common. This will explain a connection pattern for a simple feed-forward network with the nodes of ⌈ r as the set of neurons in the r-th layer. A photo of a basic version of this construction is given listed below.


We have actually given a technique to assigning chart theoretic information to sensory or constant data. It depends on the project of geometric info in the kind of a range function on the set of functions of a data set. There are lots of building and constructions based on the mathematical discipline of geography (the study of shape) that should be helpful for this task. The method ought to yield the following:

  • Improvement of the transparency of the algorithms, since the sets of neurons are now organized in understandable geometric patterns.  In particular, coloring the geometric structures by activation values of neurons will make clearer the functioning of the network.
  • Improvement  in the explainability of the outputs, based also on the coloring of the topological models of the feature sets.  What such colorings often yield is a simplified understanding of the behavior of data based on groups of features rather than individual features, as is shown in the Ayasdi blog “Machine Intelligence for Statistical Inference and Human Interpreta….”
  • Because the topological model constructions have nodes that deal with groups of features, it is to be expected that using such models will help reduce the overfitting often occurring in such models.  Preliminary experiments suggest that this does in fact occur.
  • Structures that lend themselves to topological data analysis may be classified and their topological properties might have interesting and useful interpretations. For example, it has been shown that the spatial filters of CNN’s arrange themselves in simple circles and that these are indicative of a network’s ability to generalize to unseen data. In addition, the persistent homology of the activation maps of CNN’s can be used to detect adversarial examples.
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