Applied Topology

Around the time of my recent submission Multicore homology via Mayer Vietoris an interesting optimization was made which seems to have a nice impact on the running time of the persistence algorithm.

When computing persistence  the boundary matrix is usually stored sparsely. In particular, over $$\mathbb{Z}_2$$ each column stores only the indices of the rows of the matrix which are nonzero in it’s column.

In the parlance of [ZC ’05]  each column is identified as:

$$x = \partial(\textrm{Cascade}[\sigma]).$$

The column $$x$$ is laid out in memory as a [dynamic] array of computer words  corresponding to row indices, with the natural order of these words being the filtration order.

The persistence algorithm proceeds iteratively, column by column, adding previous columns to the current one. If $$x$$ is the current column being operated on the next column we will add to it

can be identified as:

$$y = \textrm{Cascade[Youngest[}\partial(\textrm{Cascade}[\sigma])\textrm{]]]}$$

Until either the column is zeroed out or a unique pairing occurs, e.g. $$y = 0$$.

Now let me restate the definition of $$y$$ as a procedure:

Step 1) The youngest entry of $$x$$ is identified.

Step 2) The address of the column $$y$$ is computed.

Step 3) The address of the column $$y$$ is fetched into CPU registers.

Usually, this sequence of three steps occurs only after the previous addition completed. Chain addition is generally are implemented as a set_symmetric_difference from begin() to end() on each chain, e.g. oldest entry of each column to the youngest.

In recent versions of both CTL as well as PHAT a change was made to how these column additions  are carried out. The columns are added now added from youngest to oldest elements. This should have the advantage of better locality of reference.

This is because when inspecting the youngest entry we are loading it, along with some amount of the end of the chain into memory. If we then proceeded to add front to back we might end up retiring this cache line in favor of the beginning of the array.

However, adding backwards has even more benefits. Once we add backwards we know that the first pair of elements we inspect will agree, necessarily. However, it is possible,  that the two vectors we add together agree at their tail significantly more. We may delay the allocation of the output buffer for the sum until we find the first mismatch between these two chains. This saves us a small amount of memory. But their is even more.

Supposing that $$x$$ is not a positive column, e.g. after a call to mismatch we will find a pair of elements in disagreement, precisely the younger of these two elements is the new youngest element.  We may now compute the address of the next column to add to $x$ and request that the computer prefetches the column at that address.

In other words we get some added low level parallelism: as we add the pair of current columns, we are simultaneously fetching the next column to add.

How well does all of this work? In my opinion, quite well. Here is a  plot of the running time for the persistence algorithm showing the original running time, the running time when adding backwards, and the running time when adding backwards with prefetching, on the various datasets from the multicore paper

Feel free to comment and discuss on the results of this experiments. I’ll answer questions the best I can.

At this past year at the IMA, there has been some attention spent on a number of interesting aspects in bringing persistent homology closer to statistical methods.

One core step in this process has been to figure out what we mean by an average persistence diagram, a question that has an answer proposed by Munch, Bendich, Turner, Mukherjee, Mattingly, Harer and Mileyko in various constellations.

The details on Frechet means for persistent homology is not what this post is about, however. Instead, I want to bring up something I just saw presented today at ICML. Marco Cuturi and Arnaud Doucet got a paper entitled Fast Computation of Wasserstein Barycenters accepted to this large machine learning conference.

In their paper, Cuturi-Doucet present work on computing the barycenter (or average, or mean) of N probability measures using the Wasserstein distance: they articulate the transport optimization problem of finding a measure minimizing Wasserstein distance to all the N given measures, and present a smoothing approach that allows them to bring the problem onto a convex optimization shape. In the talk — though not present in the paper — the authors argue that they can achieve quadratic running times through these approximation steps. Not only that, but their approach ends up amenable for GPGPU computation and significant parallelization and vectorization benefits.

Their approach works anywhere that the Wasserstein metric is defined, and so in particular should work (and most likely give the same results) on the persistence diagram setting studied by the persistent statistician constellations mentioned above.

I for one would be excited to hear anything about the relations between these (mostly) parallel developments.

Ryan H. Lewis summarizes the morning talks, and Andrew Cooper the afternoon talks:

Christopher Hoffman gives the first talk about recent advanced in random topology. He began by discussing the stopping time for monotone graph properties. An example is for a sequence of graphs $$\langle G_j \rangle$$ we define:
$$ \tau_{\textrm{connected}} = \min j \textrm{ such that } G_j \textrm{ is connected}$$
For example in the following example we have $$\tau_{\textrm{connected}} = 4$$

He is interested in generalizing two results about monotone Erdos-Renyi random graphs to facts about erdos-renyi random simplicial complexes.

The Linial-Meshulam model is a collection $$Y_i$$ of 2-dimensional simplicial complexes where $$Y_0$$ is a complete graph on $$n$$ vertices and $$Y_i = Y_{i-1} \cup \{\textrm{a } 2 \textrm{ cell} \}.$$

It turns out that the first 2-cycle either has 4 faces with probability converging to $$c_0 = .909$$ or it is larger than $$\frac{n}{\log{(n)}}$$ with probability converging to 1 – $$c_0$$.

To generalize the second result to studying isolated edges one can study when the  $$H_1(Y, C)=0$$ for  $$\mathbb{Z}, \mathbb{Z}_2,$$ and $$\mathbb{Q}$$ coefficients, as well as the $$\pi_1(Y) = 0$$. He presents a series of results relating the stopping times for these events.

In the future they want to use probabilistic methods to demonstrate the existence of complexes desired but unobtained by classical methods in topology.

Paul Villeuotrox (sp?)

Talks about using persistent homology on a wide range of epithelial cells.

By viewing such a structure as a cover of the plane, it’s nerve is a 2D topological space. A filtration of this space is given by assigning to a vertex it’s degree, and each cell the maximum filtration value of it’s boundary.

He has found that persistent homology has proven useful for studying the structure of these cell networks.

He finds that by comparing the barcodes produced from these pipelines to the barcodes produced by complexes built on a random complexes whose underlying graph is endowed with the degree distribution that has been observed empirically, that while persistent $$H_0$$ appears to be similar between these two types of complexes, persistent $$H_1$$ seems to be very different.

Raul Rabadan talked about The Topology of Evolution.

The only figure in Darwin’s Origin of Species is a (mostly-binary) tree. This “Tree of Life” paradigm used starting in the 1970s to analyze genomic sequences. The first major discovery using the tree paradigm was Carl Woese’s 1977 discovery of Domain Archaea.

Woese excluded viruses because they lacked some of the genes he used. But even if he had included them, he would have had trouble: viruses have a high level of horizontal gene transfer, so the choice of tree as a structure to represent the phylogeny is not very good for viruses.

Nor is it very good for bacteria and archaea. Nor is it very food for plants (even Darwin knew this). Nor is it very good for us: when you get gonorrhoea, it gets you! 10% of gonorrhoea genes are human. 8% of human genes are viral. Your genome is something like a “cemetery of past infections” rather than a list of all your ancestors.

If we can’t use a tree to model evolution, what can we use? Mathematically, the problem is:

Given a set of genomes and a way of comparing them, how do we represent their relationships without importing (too many) assumptions from biology?

We would like an answer which is statistical and incorporates the notion of scale. We’d also like to detect when clonal (descent) transfer happens, and when horizontal (non-descent) transfer happens. Answer: use persistent homology to detect the topology of the genetic data!

Persistence detects not just topology, but topology at scales. In 0th homology, scale represents taxa: as we increase the filtration value, we are collecting together more and more distantly related genomes. In 1st homology, a long bar represents a transfer between distantly-related taxa.

For example, though overall flu genes show  a lot of cycles, if we restrict our attention to a particular segment there are almost no long-persisting cycles. HIV, on the other hand, has persistent cycles even when we focus on small suites of genes. Thus persistence detects the fact that gene transfer in flu occurs by trading whole segments, whereas in HIV it occurs by trading much smaller units of genetic material.

For human genomic data, persistence bars are about 2 centiMorgans-per-megabase long.

Michael Robinson talked about Morphisms between Logic Circuits

Logic circuits are described by their truth tables. But computers take time to do computations: fast input switching yields the “wrong” output (as evidenced by the flickering screen on Dr. Robinson’s slides). How can we analyze the failures of circuits due to problems of timing? Use sheaves!

Sheaves allow local specifications (we are really good at understanding small circuits) to determine global behavior (what we need to get a handle on). Plus sheaves whose stalks are vector spaces are `just’ linear-algebraic, so we can compute their cohomology using easy, well-known techniques.

As we try to associate a vector space to each logic gate, we encounter various aspects of engineering practice like one-hot encoding.

The zeroth cohomology of the switching sheaf detects the (synchronous) classical logic behavior of the system. The first cohomology of the switching sheaf detects stored information (hence, the possibility of a timing problem in the circuit).

But cohomology of the switching sheaf doesn’t tell us everything. The categorification approach says we should consider morphisms to get more information. Given a circuit we want to understand, we can construct a circuit with the same logical behavior.Then we can ask how many morphisms of the switching sheaves there are which cover the identity on inputs and outputs.

Sometimes there aren’t any such morphisms. Sometimes there are a few. Apparently there are never exactly three.

Today’s summaries are provided by Isabel Darcy.

Omer Bobrowski talked about Topological Estimation for Super Level Sets.  The goal is to determine the homology of an unknown space from a sample of noisy data points.  Super-level sets of a density function f correspond to dense regions: {x | f(x) > L}.  In general, the density function is not known but can often be estimated.  One can try to reconstruct the homology by looking at \(U_n(L, r)\) = the union of balls of fixed radius r around each point in a super level set, {x | f(x) > L}.

But if not enough points are chosen (i.e., L large), then the space may not be adequately covered by \(U_n(L, r)\).  If too many points are chosen (i.e., L small), then more noise may be picked up.  However one can obtain the correct homology with high probability by looking at how \(U_n(L_1, r)\) includes into \(U_n(L_2, r)\) for \(L_2 < L_1\).  This induces a map on their homologies.  The image of this map is the homology estimator, which equals the correct homology of the space of interest with high probability.

Elizabeth Munch talked about The Interleaving Distance for Reeb Graphs. Reeb graphs provide an efficient description to understand the properties of a real-valued function on a topological space and are useful in many applications.  Thus it would be very useful to have a method for comparing two Reeb graphs.  Interleavings (and interleaving distances) have been used to compare persistence modules.  Interleavings can be applied to Reeb graphs by defining a generalization of a Reeb graph as a functor.  One consequence is a concrete algorithm for smoothing a Reeb graph in order to remove noise.

Peter Bubenik talked about Generalized Persistence Modules and Stability.  Generalized persistence modules is an abstract formulation of persistence modules using category theory which includes many forms of persistence modules that are currently studied.  One consequence of this formulation is that one can give simpler common proofs for many standard results such as stability.

Yuliy Baryshnikov talked about Integral Operators in Euler World. One can integrate functions that take on a finite number of values using the Euler characteristic as the measure.  For example if f(x) = 4 for x in [0, 1] and 0 elsewhere, then the integral of f with respect to the Euler characteristic = 4 times the Euler characteristic of [0, 1] = 4(-1 + 2) = 4.  In applications, sometimes it is easier to solve a problem by transforming it into a simpler problem using an integral transform.

An example of an integral transform is convolution: Given functions f and g, one can create a new function, f*g, where the value f*g(x) is obtained from f and g by integrating the product f(t)g(x-t) with respect to the euler characteristic.  Given f and f*g, one would like to be able to reconstruct g: that is one would like to calculate the inverse of the Euler integral transform.  Cases where one can calculate the inverse transform were discussed.

Summaries for Day 3 are contributed by Rachael Phillips.

Jeff Erickson talked about efficiently hex-meshing things with topology. With a hex mesh, a polyhedra with six quadrilateral facets, there can be a quadrilateral mesh that can be extended to a hexahedral mesh of the interior volume. This can only happen when there are an even amount of quadrilaterals and none of the cycles are odd for the hex mesh. If such a mesh exists, then a polyhedron in 3-dimensional Euclidean space with quadrilateral facets can be constructed in polynomial time. These are extended to domains that have disconnected boundaries and are continued from Thurston, Mitchell, and Eppstein, where the odd cycle criteria is trivial. The idea is to look at a quadrilateral figure and extend that figure to the interior. So, the importance is not in the shape of that figure, since we are not looking at the geometry of this figure, but the topology.

 Jose Perea talked about Obstructions to Compatible Extensions of Mappings. From Betti numbers in 1994 to Zig-Zag persistence in 2009, there have been several classic invariants in algebraic topology. The basic ones being from a point cloud, constructing a filtration and using that filtration to compute Betti numbers, which tell us about the number of k-dimensional holes within a metric space. Instead of this, it would be useful to come up with new ways of encoding multi-scale information from data. The main goal is to be able to fit our data into a model for the best methods. Using extending sections and the retraction problem, Mumford data is used to fit the model. The question is, how far do you have to go for the model to be good? From local to global, the model tells us the death-like events, where an example would be compatible extensions. The birth-like events where the filtration of each level extends to the next. Once these models are found, it extends compatibility once the model has been extended. The main goal being to extend the previous invariant methods to new invariant methods for data analysis.

Donald Sheehy talked about Nested Dissection and (Persistent) Homology. Using nested dissection, this is a way of solving systems of linear equations. This method is an improvement of the naive Gaussian elimination. The reason that it is important to improve Gaussian elimination is because it is a long process that takes a lot of computer memory. It is a method that needs to be improved when using computers to solve it. By building a filtered simplicial complex and computing the persistent homology, we can try to speed up the process of elimination. Normally, Gaussian elimination has a running time of $$O(n^3)$$, or even worse using Strassen it is a running time of $$O(n^{\log_27})$$. Nested Dissection removes a random column in a matrix and separates the graph into two pieces. When a matrix is separated in two pieces, it improves the matrix multiplication and using topology, while doing Gaussian on the boundary. The nested dissection computes the persistent homology of the space of the matrix. Using four methods such as Mesh Filtrations, Nested Dissection, Geometric Separation and Output sensitive persistence algorithm, there is a theorem that improves the asymptotic running time of the persistence algorithm.

Shmuel Weinberger talked about Complex and Simple “Topological” invariants. It is common in the study of topological data analysis that the values of topological invariants are discussed. When you calculate homology it does not always give you enough information. The goal is not to look at the dynamics, but learn about dynamics. It is possible to look at the persistent homology of any data set, unfortunately, conditions on the noise, “invariance” with high-probability. It is important is to find examples of Probability Approximately Correct (PAC) computable ideas. Knowing when noise is a problem and when it is not is the main goal of this talk.

For day 2, Sara Kalisnik and Andrew Blumberg are giving summaries of the talks we heard:

Gunnar Carlsson: Persistence barcodes are natural invariants of finite metric spaces useful for studying point cloud data. However, they are not well adapted to standard machine learning methods. One approach to this problems is to equip the set of barcodes with a metric, but an even simpler would be to provide coordinates for the set of barcodes. Gunnar Carlsson talked about coordinatizations of barcode spaces and their properties. He also proposed some ways in which they could be used to obtain information from multidimensional persistence profiles.

Vanessa Robins focused on on-going work about applications of discrete Morse theory and persistent homology to analyzing images of porous materials.  She explained specific connections between the physical structure of the material and the patterns of persistence diagrams.  The results presented were particularly exciting insofar as they represented a very serious and thorough application of computational topology to large quantities of real data.

Radmila Sazdanovic introduced categorification and provided several examples in pure mathematics, especially knot theory. Among others, categorifications of Jones and chromatic polynomials.

Sayan Mukherjee had two distinct themes.  In the first part of the talk, he discussed results (with Boyer and Turner) on the “persistent homology transform” and “Euler characteristic transform”, which roughly speaking are invariants of an object in \(\mathbb{R}^2\) or \(\mathbb{R}^3\) obtained by taking the ensemble of persistent homology or Euler characteristic of slices (relative to some fixed orientation vector).  It turns out these are sufficient statistics and moreover seem quite successful in classification problems (e.g., for primate bones).  The second part of the talk focused on the problem of manifold learning in the context of mixtures of hyperplanes of different dimensions.  The key insight is that a Grassmanian embedding due to Conway, Hardin, and Sloane allows the use of distributions on the sphere to carry out statistical procedures on spaces of hyperplanes.

Day 1 of ATMCS 6 is now (mostly) over. Small groups of applied topologists are roaming the streets of Vancouver looking for sights, food or drink, while the less hardy of us have already eaten and retired to our rooms for a quiet night, or a night full of last-minute preparations.

Vin de Silva talked about his currently ongoing research into interesting new perspectives on persistence stability theorems and foundational models for persistence modules. One thing that really caught my attention was the idea of metric certificates: many metrics are defined as the supremum or infimum over a range of potential comparison points. The certificates idea summarizes all these approaches under a common header – a comparison point is a certificate, and the metric is produced by optimizing across certificates.

This is put to use to produce extension theorems of the form
If \(A\) is a subspace of \(B\), and there is a 1-Lipschitz map \(A \to M\), then we can construct a 1-Lipschitz map \(B \to M\).
These theorems turn out to hold in a bunch of situations, and to be highly relevant for persistent homology.

Amit Patel talked about his work on Quillen 2-categories and their relationship to persistent homology. It turns out there are ways of talking about persistent homology that pull in some sheaf-theoretic perspectives and naturally produce a Quillen 2-category that encodes much of the structure.

Sarah Day talked about Conley index theory and symbolic dynamics; with some research geared towards using symbolic dynamics approximations of dynamical systems to discover models and pick out cycles and stable behaviors. Within this project, Conley indices turn out to be useful tools.

Tamal Dey talked about the Graph induced complex, and work with Fengtao Fan and Yusu Wang on data sparsification by building topological models and simplifying the computational complexity of generating topological inferences.

I’ll see if I can recruit volunteers from the audience to keep a stream of conference updates flowing here.