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1. Fast Approximate Counting of Cycles

   https://doi.org/10.4230/LIPIcs.ICALP.2024.37  

   Censor-Hillel, Keren; Even, Tomer; Vassilevska_Williams, Virginia (January 2024, Schloss Dagstuhl – Leibniz-Zentrum für Informatik)

   Bringmann, Karl; Grohe, Martin; Puppis, Gabriele; Svensson, Ola (Ed.)

   We consider the problem of approximate counting of triangles and longer fixed length cycles in directed graphs. For triangles, Tětek [ICALP'22] gave an algorithm that returns a (1±ε)-approximation in Õ(n^ω/t^{ω-2}) time, where t is the unknown number of triangles in the given n node graph and ω < 2.372 is the matrix multiplication exponent. We obtain an improved algorithm whose running time is, within polylogarithmic factors the same as that for multiplying an n× n/t matrix by an n/t × n matrix. We then extend our framework to obtain the first nontrivial (1± ε)-approximation algorithms for the number of h-cycles in a graph, for any constant h ≥ 3. Our running time is Õ(MM(n,n/t^{1/(h-2)},n)), the time to multiply n × n/(t^{1/(h-2)}) by n/(t^{1/(h-2)) × n matrices. Finally, we show that under popular fine-grained hypotheses, this running time is optimal. 

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2. Near-Optimal Derandomization of Medium-Width Branching Programs

   https://doi.org/10.1145/3564246.3585108  

   Putterman, Aaron; Edward Pyne (June 2023, 55th Annual ACM Symposium on Theory of Computing, STOC 2023)

   We give a deterministic white-box algorithm to estimate the expectation of a read-once branching program of length n and width w in space Õ(logn+√logn·logw). In particular, we obtain an almost optimal space Õ(logn) derandomization of programs up to width w=2√logn. Previously, the best known space complexity for this problem was O(min{logn· logw,log3/2n+√logn· logw}) via the classic algorithms of Savitch (JCSS 1970) and Saks and Zhou (JCSS 1999), which only achieve space Õ(logn) for w=polylog(n). We prove this result by showing that a variant of the Saks-Zhou algorithm developed by Cohen, Doron, and Sberlo (ECCC 2022) still works without executing one of the steps in the algorithm, the so-called random shift step. This allows us to extend their algorithm from computing the nth power of a w× w stochastic matrix to multiplying n distinct w× w stochastic matrices with no degradation in space consumption. In the regime where w≥ n, we also show that our approach can achieve parameters matching those of the original Saks-Zhou algorithm (with no loglog factors). Finally, we show that for w≤ 2√logn, an algorithm even simpler than our algorithm and that of Saks and Zhou achieves space O(log3/2 n). 

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3. Quantum SDP Solvers: Large Speed-Ups, Optimality, and Applications to Quantum Learning

   Brand; Kalev, Amir; Li, Tongyang; Lin, Cedric Yen-Yu; Svore, Krysta M.; Wu, Xiaodi (July 2019, Leibniz international proceedings in informatics)

   We give two new quantum algorithms for solving semidefinite programs (SDPs) providing quantum speed-ups. We consider SDP instances with m constraint matrices, each of dimension n, rank at most r, and sparsity s. The first algorithm assumes an input model where one is given access to an oracle to the entries of the matrices at unit cost. We show that it has run time O~(s^2 (sqrt{m} epsilon^{-10} + sqrt{n} epsilon^{-12})), with epsilon the error of the solution. This gives an optimal dependence in terms of m, n and quadratic improvement over previous quantum algorithms (when m ~~ n). The second algorithm assumes a fully quantum input model in which the input matrices are given as quantum states. We show that its run time is O~(sqrt{m}+poly(r))*poly(log m,log n,B,epsilon^{-1}), with B an upper bound on the trace-norm of all input matrices. In particular the complexity depends only polylogarithmically in n and polynomially in r. We apply the second SDP solver to learn a good description of a quantum state with respect to a set of measurements: Given m measurements and a supply of copies of an unknown state rho with rank at most r, we show we can find in time sqrt{m}*poly(log m,log n,r,epsilon^{-1}) a description of the state as a quantum circuit preparing a density matrix which has the same expectation values as rho on the m measurements, up to error epsilon. The density matrix obtained is an approximation to the maximum entropy state consistent with the measurement data considered in Jaynes' principle from statistical mechanics. As in previous work, we obtain our algorithm by "quantizing" classical SDP solvers based on the matrix multiplicative weight update method. One of our main technical contributions is a quantum Gibbs state sampler for low-rank Hamiltonians, given quantum states encoding these Hamiltonians, with a poly-logarithmic dependence on its dimension, which is based on ideas developed in quantum principal component analysis. We also develop a "fast" quantum OR lemma with a quadratic improvement in gate complexity over the construction of Harrow et al. [Harrow et al., 2017]. We believe both techniques might be of independent interest. 

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4. Generating Cryptographically-Strong Random Lattice Bases and Recognizing Rotations of Zn

   Blanks, Tamar Lichter; Miller, Stephen D. (January 2021, Lecture notes in computer science)

   Cheon, Jung Hee; Tillich, Jean-Pierre (Ed.)

   Lattice-based cryptography relies on generating random bases which are difficult to fully reduce. Given a lattice basis (such as the private basis for a cryptosystem), all other bases are related by multiplication by matrices in GL(n,Z). We compare the strengths of various methods to sample random elements of GL(n,Z), finding some are stronger than others with respect to the problem of recognizing rotations of the Zn lattice. In particular, the standard algorithm of multiplying unipotent generators together (as implemented in Magma’s RandomSLnZ command) generates instances of this last problem which can be efficiently broken, even in dimensions nearing 1,500. Likewise, we find that the random basis generation method in one of the NIST Post-Quantum Cryptography competition submissions (DRS) generates instances which can be efficiently broken, even at its 256-bit security settings. Other random basis generation algorithms (some older, some newer) are described which appear to be much stronger. 

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5. Fast Regression for Structured Inputs

   Meyer, Raphael; Musco, Cameron; Musco, Christopher; Woodruff, David P.; Zhou, Samson (April 2022, International Conference on Learning Representations (ICLR))

   We study the $$\ell_p$$ regression problem, which requires finding $$\mathbf{x}\in\mathbb R^{d}$$ that minimizes $$\|\mathbf{A}\mathbf{x}-\mathbf{b}\|_p$$ for a matrix $$\mathbf{A}\in\mathbb R^{n \times d}$$ and response vector $$\mathbf{b}\in\mathbb R^{n}$$. There has been recent interest in developing subsampling methods for this problem that can outperform standard techniques when $$n$$ is very large. However, all known subsampling approaches have run time that depends exponentially on $$p$$, typically, $$d^{\mathcal{O}(p)}$$, which can be prohibitively expensive. We improve on this work by showing that for a large class of common \emph{structured matrices}, such as combinations of low-rank matrices, sparse matrices, and Vandermonde matrices, there are subsampling based methods for $$\ell_p$$ regression that depend polynomially on $$p$$. For example, we give an algorithm for $$\ell_p$$ regression on Vandermonde matrices that runs in time $$\mathcal{O}(n\log^3 n+(dp^2)^{0.5+\omega}\cdot\text{polylog}\,n)$$, where $$\omega$$ is the exponent of matrix multiplication. The polynomial dependence on $$p$$ crucially allows our algorithms to extend naturally to efficient algorithms for $$\ell_\infty$$ regression, via approximation of $$\ell_\infty$$ by $$\ell_{\mathcal{O}(\log n)}$$. Of practical interest, we also develop a new subsampling algorithm for $$\ell_p$$ regression for arbitrary matrices, which is simpler than previous approaches for $$p \ge 4$$. 

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