Risi Kondor

Life and career

Risi Kondor came into machine learning by way of physics and pure mathematics, and he never really left either behind. Hungarian by birth, he trained in Europe before crossing the Atlantic for doctoral work at Columbia University, where he wrote a dissertation on group-theoretic methods in machine learning — a title that in the mid-2000s read as eccentric and today reads as prophetic. At the time, the field's center of gravity was kernel methods and, shortly after, the first stirrings of deep learning. Kondor was interested in something orthogonal to both: the question of what a learning algorithm should look like when you already know, in advance, that the world it is modeling has symmetries.

His early work was done in the orbit of John Lafferty at Carnegie Mellon, and the collaboration produced one of the most-cited ideas of the kernel era — diffusion kernels, which extended the machinery of support vector machines from vectors in Euclidean space to nodes in a graph. He then moved through postdoctoral positions in Europe and the United States, including time in the mathematics-of-information community at Caltech, before settling at the University of Chicago, where he holds a joint appointment across computer science and statistics.

Chicago suited him. It is a department comfortable with mathematics for its own sake, and Kondor spent the 2010s building a research program around a single conviction that was, for most of that decade, unfashionable: that the right way to get a neural network to respect the symmetries of its data is not to show it enough examples until it figures them out, but to build the symmetry into the architecture as a hard constraint. Rotate a molecule and its predicted energy should not change at all — not approximately, not on average over a large training set, but exactly, as a matter of algebra. He wrote software (including tooling for fast Fourier transforms over the symmetric group, a genuinely difficult computational problem) as well as theory, and he trained a generation of students who carried equivariant methods into chemistry, physics, and molecular biology.

The vindication came in the late 2010s and 2020s. Equivariant neural networks became a mainstream architecture class, indispensable in computational chemistry and materials science, and the geometric deep learning program absorbed much of the vocabulary Kondor had helped formalize. He remains an unusual figure in modern ML: a researcher whose papers are more likely to contain a representation-theoretic decomposition than a benchmark table.

Key contributions

**Diffusion kernels on graphs.** With Lafferty, Kondor introduced a principled way to define a positive-definite kernel on discrete structures. The construction is elegant: take the graph Laplacian, exponentiate it, and you get a kernel matrix that is the discrete analogue of the Gaussian kernel — the heat equation run for a fixed time, measuring similarity by how much probability mass diffuses from one node to another. This gave kernel machines a rigorous entry point into relational data, and it remains a foundational reference for anyone working on graph-based semi-supervised learning, where the diffusion operator is precisely what propagates a handful of labels across a large unlabeled manifold.

**Harmonic analysis on the symmetric group.** Much of Kondor's most technically demanding work concerns Fourier analysis over non-commutative groups, especially the group of permutations. If your data is a graph and you want a representation that does not depend on how you happened to number the nodes, you are asking a question about functions on the symmetric group, and the natural tool is the representation theory of that group. Kondor developed both the theory and the fast algorithms — including graph invariants built from the higher-order Fourier coefficients of a graph's adjacency structure, which capture strictly more than the eigenvalue spectrum alone, and applications to hard combinatorial problems such as quadratic assignment and identity management in multi-object tracking.

**Equivariance theory for neural networks.** Kondor's most influential theoretical result, developed with Shubhendu Trivedi, concerns what happens when you demand that a neural network's layers commute with the action of a compact group. The answer is strikingly rigid: under natural conditions, any equivariant linear layer must be a generalized convolution over that group. Convolution, in other words, is not an engineering choice; it is forced. This reframed the entire discussion — CNNs are translation-equivariant because convolution is the only translation-equivariant option, and the way to build networks for rotations, permutations, or the sphere is to identify the corresponding group and construct the corresponding convolution.

**Spherical and molecular architectures.** He put this theory to work. His group's spherical networks operate entirely in Fourier space, using Clebsch–Gordan coefficients to combine representations nonlinearly without ever leaving the frequency domain — solving the problem that pointwise nonlinearities destroy equivariance on curved domains. On graphs, his covariant compositional networks generalized message passing so that node aggregations transform as tensors rather than collapsing to permutation-invariant sums, addressing head-on the known expressiveness ceiling that limits standard message-passing networks to what the Weisfeiler–Lehman isomorphism test can distinguish. The molecular architectures that followed treat rotational equivariance as an exact structural property of the network.

**Multiresolution matrix factorization.** A quieter but characteristic contribution: rather than approximating a large matrix by a low-rank factorization, Kondor and collaborators proposed factoring it into a hierarchy of sparse orthogonal transforms — a wavelet-like decomposition for matrices, appropriate when structure exists at many scales rather than concentrating in a few dominant directions.

In battle

Kondor is the sharpest specialist on the roster, and the numbers say so bluntly. He carries 103 problems with a mean score of 22.3 and a median of 18 — but a range that tops out at 98. Sixty-two of his problems score at or below 20. This is not a generalist who is merely good everywhere; it is a scalpel, devastating on a narrow class of problems and close to inert elsewhere.

Where he wins, he wins outright. His single best card is **P272, "Learning on the molecule graph"** (98) — as close to a perfect match as this roster produces. That problem asks for a representation invariant to arbitrary node numbering, equivariant where geometry demands it, and validated against graph-isomorphism expressiveness limits. That is not adjacent to Kondor's program; it *is* his program, from the graph kernels of the early 2000s through covariant compositional networks. **P096, "Tensors don't factor like matrices"** (80) plays to the same core: multilinear structure and representation theory are his native language in a way they are not for most of the field. His `networks` category average of 81.5 is the highest single-category figure he posts.

Below that peak, his strength decays into competence. **P027, "Classify with 30 labels and 30,000 unlabeled"** (69) is diffusion-kernel territory — semi-supervised learning on a graph is what the heat kernel was built for. **P273, "Cut the image, weight the graph"** (65) rewards his spectral-graph fluency. **P025, "Sixty thousand digits"** (56), **P023, "The line between two clouds"** (55), and **P024, "XOR"** (47) are the generic mid-table results of a strong ML researcher on canonical problems he did not personally invent — respectable, rarely decisive.

The losses are instructive, and a student should not be surprised by them. Kondor collapses on classical statistics and on anything requiring probability estimation from counts. **P144, "The spam in the inbox"** (4) — naive Bayes with Laplace smoothing — and **P143, "The coefficient that flips sign"** (4) — multicollinearity in linear regression — sit at the very bottom, and his `regression` (16.4) and `testing` (10.0) category averages confirm the pattern. **P141, "Why least squares, exactly?"** (6) asks for Gauss–Markov reasoning he simply does not carry. **P145, "Just look at the neighbors"** (5) is almost comic: nearest neighbors is the one method that needs no structural theory at all, which is exactly why he adds nothing to it.

He is equally helpless in reinforcement learning and games. **P200, "Beat the world champion"** and **P201, "The dice make it learnable"** both score 5, and his `games` average of 5.0 is the worst on his card. Sequential decision-making under stochastic dynamics has no group to exploit.

Play Kondor when the problem statement mentions molecules, graphs, spheres, rotations, permutations, or invariance. Bench him the moment the data is a flat table of numbers.