Higher Order Random Walks, Local Spectral Expansion, and Applications

The study of spectral expansion of graphs and expander graphs has been an extremely fruitful line of research in Mathematics and Computer Science, with applications ranging from random walks and fast sampling to optimization. In this dissertation, we study high dimensional local spectral expansion, which is a generalization of the theory of spectral expansion of graphs, to simplicial complexes. We study two random walks on simplicial complexes, which we call the down-up walk, which captures a wide array of natural random walks which can be used to sample random combinatorial objects via the so-called heat-bath dynamics, and the swap walk, which can be thought as a random walk on a sparse version of the Kneser graph. First, we give a sharp bound for the spectral gap of the down-up walks in terms of the local spectral expansion. Using this bound, we argue that the natural Markov chains for (i) sampling a random independent of fixed size s of a graph G = (V,E) is rapidly mixing, so long as s ≤ |V|/(∆+η) – where ∆ is the maximum degree of any vertex in G, and η is the magnitude of the least eigenvalue of the adjacency matrix of G; and (ii) sampling a common independent set from two partition matroids of fixed size s is rapidly mixing, so long as s ≤ r/3 – where r is the maximum size of any common independent set contained in both partition matroids. Next, we study the spectrum of the swap walks, and show that using local spectral expansion we can relate the spectrum of the swap walk on any simplicial complex to the spectrum of the Kneser graph. We will mention applications of this result in (i) approximating constraint satisfaction problems (CSPs) on instances where the constraint hypergraph is a high dimensional local spectral expander; and in (ii) the construction of new families of list decodable codes based on (sparse) Ramanujan complexes of Lubotzky, Samuels, and Vishne

LIPIcs, Volume 261, ICALP 2023, Complete Volume

LIPIcs, Volume 261, ICALP 2023, Complete Volum

Quantum meets optimization and machine learning

With the advent of the quantum era, what role the quantum computer will play in optimization and machine learning becomes a natural and salient question. The development of novel quantum computing techniques is essential to showcase the quantum advantage in these fields. At the same time, new findings in classical optimization and machine learning algorithms also have the potential to stimulate quantum computing research. In the dissertation, we explore the fascinating connections between quantum computing, optimization, and machine learning, paving the way for transformative advances in all three fields. First, on the quantum side, we present efficient quantum algorithms for fundamental problems in sampling, optimization, and quantum physics. Our results highlight the practical advantages of quantum computing in these fields. In addition, we introduce new approaches to quantum complexity theory for characterizing the quantum hardness of optimization and machine learning problems. Second, on the optimization side, we improve the efficiency of the state-of-the-art classical algorithms for solving semi-definite programming (SDP), Fourier sensing, dynamic distance estimation, etc. Our classical results are closely intertwined with quantum computing. Some of them serve as stepping stones to new quantum algorithms, while others are motivated by quantum applications or inspired by quantum techniques. Third, on the machine learning side, we develop fast classical and quantum algorithms for training over-parameterized neural networks with provable guarantees of convergence and generalization. Furthermore, we contribute to the security aspect of machine learning by theoretically investigating some potential approaches to (classically) protect private data in collaborative machine learning and to (quantumly) protect the copyright of machine learning models. Fourth, we investigate the concentration and discrepancy properties of hyperbolic polynomials and higher-order random walks, which could potentially be applied to quantum computing, optimization, and machine learning.Computer Science

Edge Isoperimetry and Rapid Mixing on Matroids and Geometric Markov Chains

We show how to bound the mixing time and log-Sobolev constants of Markov chains by bounding the edge-isoperimetry of their underlying graphs. To do this we use two recent techniques, one involving Average Conductance and the other log-Sobolev constants. We show a sort of strong conductance bound on a family of geometric Markov chains, give improved bounds for the mixing time of a Markov chain on balanced matroids, and in both cases nd lower bounds on the logSobolev constants of these chains

Edge Isoperimetry and Rapid Mixing on Matroids and Geometric Markov Chains

We show how to bound the mixing time and log-Sobolev constants of Markov chains by bounding the edge-isoperimetry of their underlying graphs. To do this we use two recent techniques, one involving Average Conductance and the other log-Sobolev constants. We show a sort of strong conductance bound on a family of geometric Markov chains, give improved bounds for the mixing time of a Markov chain on balanced matroids, and in both cases find lower bounds on the log-Sobolev constants of these chains. 1

LIPIcs, Volume 251, ITCS 2023, Complete Volume

LIPIcs, Volume 251, ITCS 2023, Complete Volum