18 research outputs found
Sign and Basis Invariant Networks for Spectral Graph Representation Learning
Many machine learning tasks involve processing eigenvectors derived from
data. Especially valuable are Laplacian eigenvectors, which capture useful
structural information about graphs and other geometric objects. However,
ambiguities arise when computing eigenvectors: for each eigenvector , the
sign flipped is also an eigenvector. More generally, higher dimensional
eigenspaces contain infinitely many choices of basis eigenvectors. These
ambiguities make it a challenge to process eigenvectors and eigenspaces in a
consistent way. In this work we introduce SignNet and BasisNet -- new neural
architectures that are invariant to all requisite symmetries and hence process
collections of eigenspaces in a principled manner. Our networks are universal,
i.e., they can approximate any continuous function of eigenvectors with the
proper invariances. They are also theoretically strong for graph representation
learning -- they can approximate any spectral graph convolution, can compute
spectral invariants that go beyond message passing neural networks, and can
provably simulate previously proposed graph positional encodings. Experiments
show the strength of our networks for molecular graph regression, learning
expressive graph representations, and learning implicit neural representations
on triangle meshes. Our code is available at
https://github.com/cptq/SignNet-BasisNet .Comment: 35 page
Notes on Randomized Algorithms
Lecture notes for the Yale Computer Science course CPSC 469/569 Randomized
Algorithms. Suitable for use as a supplementary text for an introductory
graduate or advanced undergraduate course on randomized algorithms. Discusses
tools from probability theory, including random variables and expectations,
union bound arguments, concentration bounds, applications of martingales and
Markov chains, and the Lov\'asz Local Lemma. Algorithmic topics include
analysis of classic randomized algorithms such as Quicksort and Hoare's FIND,
randomized tree data structures, hashing, Markov chain Monte Carlo sampling,
randomized approximate counting, derandomization, quantum computing, and some
examples of randomized distributed algorithms