## Tuesday, May 19, 2015

### ITA, or a conference I really enjoy.

Continuing my thoughts on the STOC 2017 reboot, I went back to Boaz's original question:

What would make you more likely to go to STOC?

And thought I'd answer it by mentioning an event that I really enjoy attending. I didn't post it as a comment because it's a little out of scope for the blog post itself: it doesn't make concrete recommendations so much as relay anecdotal evidence.

The Information Theory and Applications workshop is a workshop: it doesn't have printed proceedings, and it encourages people to present work that has been published (or is under review) elsewhere. Keep that caveat in mind: the structure here might not work for a peer-reviewed venue like STOC.

Having said that, the ITA is a wonderful event to go to.
• It's in San Diego every year in February - what's not to like about that
• It runs for 5 days, so is quite long. But the topics covered change over the course of the 5 days: the early days are heavy on information theory and signal processing, and the algorithms/ml/stats shows up later in the week.
• There are multiple parallel sessions: usually 5. And lots of talks (no posters)
• There are lots of fun activities. There's an irreverent streak running through the entire event, starting with the countdown clock to the invitations, the comedy show where professional comedians come and make fun of us :), various other goofy events interspersed with the workshop, and tee-shirts and mugs with your name and picture on them.
The talks are very relaxed, probably precisely because there isn't a sense of "I must prove my worth because my paper got accepted here". Talk quality varies as always, but the average quality is surprisingly high, possibly also because it's by invitation.

But the attendance is very high. I think the last time I attended there were well over 600 people, drawn from stats, math, CS, and EE. This had the classic feel of a 'destination workshop' that STOC wants to emulate. People came to share their work and listen to others, and there was lots of space for downtime discussions.

My assertion is that the decoupling of presentation from publication (i.e the classical workshop nature of ITA), makes for more fun talks, because people aren't trying to prove a theorem from the paper and feel the freedom to be more expansive in their talks (maybe covering related results, or giving some larger perspective).

Obviously this would be hard to do at STOC. But I think the suggestions involving posters are one way of getting to this: namely, that you get a pat on the back for producing quality research via a CV bullet ("published at STOC") and an opportunity to share your work (the poster). But giving a talk is a privilege (you're occupying people's time for slice of a day), not a right, and that has to be earned.

I also think that a commenter (John) makes a good point when they ask "Who's the audience?". I'm at a point where I don't really enjoy 20 minutes of a dry technical talk and I prefer talks with intuition and connections (partly because I can fill in details myself, and partly because I know I'll read the details later if I really care). I don't know if my view is shared by everyone, especially grad students who have the stamina and the inclination to sit through hours of very technical presentations.

## Monday, May 18, 2015

### STOC 2017 as a theory festival

Over on Windows on Theory, there's a solid discussion going on about possible changes to the format for STOC 2017 to make it more of a 'theory festival'. As Michael Mitzenmacher exhorts, please do go and comment there: this is a great chance to influence the form of our major conferences, and you can't make a change (or complain about the lack of change) if you're not willing to chime in.

I posted my two comment there, and you should go and read number one  and number two. Two things that I wanted to pull out and post here are in the form of a 'meta-suggestion':
1. Promise to persist with the change for a few years. Any kind of change takes time to get used to, and every change feels weird and crazy till you get used to it, after which point it’s quite natural.
Case in point: STOC experimented one year with a two-tier committee, but there was no commitment to stick to the change for a few years, and I’m not sure what we learned at all from one data point (insert joke about theorists not knowing how to run experiments).
Another case in point: I’m really happy about the continued persistence with workshops/tutorials. It’s slowly becoming a standard part of STOC/FOCS, and that’s great.
2. Make a concerted effort to collect data about the changes. Generate surveys, and get people to answer them (not as hard as one might think). Collect data over a few years, and then put it all together to see how the community feels. In any discussion (including this one right here), there are always a few people with strong opinions who speak up, and the vast silent majority doesn’t really chip in. But surveys will reach a larger crowd, especially people who might be uncomfortable engaging in public.

## Friday, May 15, 2015

### Higher order interactions and SDM 2015

This year I'm one of the PC Chairs for SIAM Data Mining (along with Jieping Ye), and so I've been spending time in decidedly-not-sunny Vancouver. Being a PC Chair, or even being on a PC, is an exercise in constant deja vu: I hear a talk and think "Where have I heard this before" before realizing that I've probably reviewed the paper, or looked at its reviews or meta-reviews.

Being the PC chair means though that I can float around the conference freely without feeling the pressure to attend talks, network or otherwise be social: I've earned my keep :). Following standard conference networking maxims though, I made an effort to meet at least one coffee shop that I've met before and introduce myself to at least one new coffee shop, and another.

But I am attending talks ! And I enjoyed listening to some work on tensor spectral clustering by Benson, Gleich and Leskovec. It got me thinking about the larger issue of modeling higher-order interactions and what appear to be many different ways of modeling the problem.

#### The problem.

Imagine that you have a number of interacting entities. These could be points in a space, or vertices in a graph, or even dimensions of a point (aka variables). The easiest way to model a collection of such entities is to assume they're independent of each other. For example, I might draw i.i.d samples from a distribution. Or I might be looking at a collection of independent features describing an object, and so on.

Independence assumptions are powerful. They allow us to "factorize" structure into combinations of individual elements, which either leads to simple summations directly, or eventually after a log transformation of a product. This means we can deal with entities independently, and the "inference complexity blowup" is linear in the number of entities. A good example of this is the Naive Bayes approach to learning, where assuming all entities are independent leads to a likelihood cost function that's just a sum of terms, one for each entity.

I'm necessarily being rather loose with my statements here. Making them precise is possible in specific contexts, but it gets unwieldy.

But independence is too restrictive an assumption. It limits modeling power, and therefore will be unable to capture interactions that might make understanding structure a lot easier. For one thing, you'd never find correlations if you assumed that all features are independent.

#### Graphs.

The easiest form of interaction is a pairwise interaction. Modeling pairwise interactions gets us to a graph, and who doesn't love a graph ! More importantly for what follows,

a graph and its associated structures is a rich representation of a system of pairwise interactions

in that we have a colorful vocabulary and an arsenal of algorithms for talking about pairwise interactions and structures built on them.

Of course we've paid a price - in complexity. Instead of the linear cost incurred by independent entities, we now have quadratically many potential pairwise interactions to model. But (and here's the key), we can interpret a sparse graph as capturing weak interactions, and it's still a rich model to model different phenomena.

#### Higher-order interactions.

But what happens if we want to model interactions that aren't just pairwise ? What is the correct higher-order structure to model such interactions as effectively as graphs ? It turns out that there are many different ways to do this, and they all can be reduced to a sentence (pace Saunders MacLane) of the form

A graph is just a very special kind of X
for different values of X.

#### 1. The graphical model view.

A graph is just a special kind of clique intersection structure, if you only have 2-cliques.

One way to manage a collection of higher order interactions is to factorize them in a more general way. This is the basis for the clique-tree idea in graphical models, where the interaction structure is viewed as a set of complete joint interactions (aka cliques) all connected together in a tree structure for easy inference. Another name for this would be the 'bounded treewidth' model, but it misses the fact that we are allowing higher-order interactions, but in a controlled way.

The advantage of this representation is that in a true parametrized complexity way, it isolates where the true complexity is coming from (the size of each clique) from the overall complexity (the size of the graph).

#### A spectral perspective.

When graphs arise from natural sources of data (social networks and the like), we have to deal with noise and spurious signals. And the simple language of connectivity and paths is no longer robust enough. For example a graph might be connected, but only because there's one edge connecting two huge components. If this edge was spurious, we've just made a huge mistake on modeling this graph structure.

Spectral methods are currently our best way of dealing with noisy interactions. By focusing not on the topological structure of connectivity but on the amount of connectivity measured via cuts, spectral analysis of graphs has become perhaps the best way of finding structures in large graphs.

The spectral lens sees a graph through random walks on the edges. This is great for modeling a collection of pairwise interactions between entities, but not for modeling interactions among sets of entities. We have to be careful here. Spectral methods are actually quite good at finding community structure in graphs (i.e a partition into sets of vertices). What they can't do is find higher order partitionings in graphs (i.e sets of triangles or sets of special 4-vertex structures). And that's where the next three higher-order methods enter the picture

#### 2. The algebraic topology view.

A graph is just the 1-skeleton of a simplicial complex.

If we're looking to model higher-order interactions, we need a language for describing a collection of well-defined higher order structures. That's what a simplicial complex is. I'll skip the formal definition, but the basic idea is that if you have a simplex (an interacting group of entities), then all subsets must be simplices as well. But you declare the simplices first, which means that the simplex $\{a,b,c\}$ is different from the three simplices $\{a,b\}, \{b, c\}, \{c, a\}$, even though the first must contain the second.

A simplicial complex is a topological object. It generalizes a graph because a graph is what you get if you limit yourself to simplices of size at most 2. Because it's a discrete topological object, you can now play with it using all the tools of topology, and in particular very powerful tools like homology and homotopy that reveal all kinds of hidden structure not accessible via a graph metaphor.

While simplicial complexes allow you to express higher order interactions (tunnels! holes!), they don't remove the problem of noise: one edge/simplex can still change the structure in nontrivial ways. There are two approaches that researchers have taken to this problem: one spectral, and one not.

The non-spectral approach is by far the most well-established one. It is based on the idea of persistence:  a way to determine from the homology groups of the simplicial complex what structures are interesting and which ones are not. Persistence is the dominant weapon in the arsenal of topological data analysis, and I'll say no more about it here, so as to keep focus on spectral methods.

The spectral approach is less well developed, but is quite interesting. The idea is to generalize the notion of expansion from a graph to higher-order simplices, as well as generalizing the Laplacian operator to higher order simplices (or homology groups). Then a random walk on the simplicial complex can be linked to the Laplacian operator, and the eigenstructure of the operator can be linked to the existence (or nonexistence) of certain homology groups. Two places to start with this topic are one on capturing generalizations of edge expansion, and another on building random walks on simplicial complexes and connecting them to a combinatorial Laplacian.

#### 3. The differential geometry view.

A graph is just a discrete approximation of (scalar functions on) a manifold.
The graph Laplacian is a discrete approximation of the Laplacian second order differential operator, and more generally the Laplace-Beltrami operator on a manifold. Indeed, one way to build intuition for what the graph Laplacian means is that it's capturing heat diffusion on an implicit manifold that the graph is merely approximating.

The Laplace-Beltrami operator is a "zeroth-order" operator in that it applies to the zero-dimensional entities on a manifold: namely, scalar fields over the points of the manifold. Suppose you were to build a vector field instead over the manifold, and wished to reason about it. Then the generalization of the L-B operator that you'd need is called the Laplace-de Rham operator which formally acts like a Laplacian on the higher order differential forms defined over the manifold (formally, on sections of the tangent bundle). Writing down the L-R operator is a little tricky: it involves a combination of the exterior derivative and its dual (via the Hodge * operator). But one useful observation is that the L-R operator on graphs amounts to a Laplacian on the set of edges, rather than vertices.

This means that you can now treat edges as first-class objects for grouping, rather than vertices. And this is useful for higher-order clustering. Whether this can be generalized even further remains to be seen.

#### 4. The linear algebraic view.

A graph (adjacency matrix) is just a special case of a tensor structure on the entities.

This is perhaps the most well-known of the different higher-order approaches to modeling interactions, and is making the most waves right now. The idea is very simple. If we think of a graph in terms of its adjacency matrix, then each entry encodes a relation between two basic entities. If we wished to encode relations between (say) three entities, then we need a "3D matrix", or more precisely, a tensor

Of course a tensor is more than just a way to assemble a collection of triples into a box, just like a matrix is much more than just a grid of numbers. The most basic question in tensor modeling is factorization: just like we can use the SVD to write down a matrix as a linear combination of outer products of basis vectors, can we write a tensor as 3-way wedge product of basic vectors ? If so, then we've been able to identify the key 3-way factors controlling an interaction.

Tensor factorization is a hard problem: unlike the SVD, most formulations of tensor factorization are NP-hard. I won't get into this very rich topic right now, and instead will point you to some slides by Ravi Kannan as well as an older paper of his

But can we cluster using tensors ? Or more generally, is there a spectral theory of tensors, and maybe even the analog of Cheeger's inequality ? It turns out that it's possible to define eigenvalues and eigenvectors of tensors, and at least some of the standard spectral theory can be made to carry over - for more on this see these slides by Lek-Heng Lim. But we're still looking for a theory can truly work on hypergraphs or tensors in general. In the meantime, tensor-based approaches to clustering boil down to factorization and PCA-like methods, of which there are many.

#### 5. Coda

Are these approaches in any sense equivalent ? It's hard to say in general, although for a particular problem there might be connections. Certainly, the de Rham cohomology yields a nice filtration over homology groups that could be wrestled into a persistence framework (research question !!!) thus allowing us to extract robust higher-order structures from both geometric and topological considerations. The tensor approach is closely linked to probabilistic models, not surprisingly. But whether the spectral perspective (and all its attendant intuition and value) will extend nicely across these different frameworks remains to be seen.

## Tuesday, May 12, 2015

### Special Issue of Internet Mathematics on Evolving Networks

Aris Gionis and I are co-editing a special issue of Internet Mathematics on evolving networks.
The widespread adoption of digitization has opened the way to gathering large amounts of data that record detailed information for many systems of interest. Examples include telecommunication networks, online social-media platforms, and biological systems. Many of these systems are typically represented as networks, and graph-theoretic techniques are used to analyze the available data. Furthermore, as our data-gathering capacity has increased, it is now possible to collect data the record not only a static aggregate view of the underlying network, but a continuous stream of events that captures the full dynamic behavior of the network. Such events may take the form of structural changes, or they may encode different types of actions and interactions performed by the network entities. This view of time-evolving networks poses new challenges and opens new research directions. The objective is to develop the theoretical foundations and to design the algorithmic principles that will allow to efficiently manage and analyze such evolving networks.

Read more about it here.  Submissions are due Oct 15, 2015.