6 Search Results for "Rudelson, Mark"

Document
RANDOM
DOI: 10.4230/LIPIcs.APPROX/RANDOM.2025.30
Algorithmic Contiguity from Low-Degree Conjecture and Applications in Correlated Random Graphs

Authors: Zhangsong Li

Published in: LIPIcs, Volume 353, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2025)

Abstract
In this paper, assuming a natural strengthening of the low-degree conjecture, we provide evidence of computational hardness for two problems: (1) the (partial) matching recovery problem in the sparse correlated Erdős-Rényi graphs G(n,q;ρ) when the edge-density q = n^{-1+o(1)} and the correlation ρ < √{α} lies below the Otter’s threshold, this resolves a remaining problem in [Jian Ding et al., 2023]; (2) the detection problem between a pair of correlated sparse stochastic block model S(n,λ/n;k,ε;s) and a pair of independent stochastic block models S(n,λs/n;k,ε) when ε² λ s < 1 lies below the Kesten-Stigum (KS) threshold and s < √α lies below the Otter’s threshold, this resolves a remaining problem in [Guanyi Chen et al., 2024]. One of the main ingredient in our proof is to derive certain forms of algorithmic contiguity between two probability measures based on bounds on their low-degree advantage. To be more precise, consider the high-dimensional hypothesis testing problem between two probability measures ℙ and ℚ based on the sample Y. We show that if the low-degree advantage Adv_{≤D}(dℙ/dℚ) = O(1), then (assuming the low-degree conjecture) there is no efficient algorithm A such that ℚ(A(Y) = 0) = 1-o(1) and ℙ(A(Y) = 1) = Ω(1). This framework provides a useful tool for performing reductions between different inference tasks.
Cite as

Zhangsong Li. Algorithmic Contiguity from Low-Degree Conjecture and Applications in Correlated Random Graphs. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 353, pp. 30:1-30:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{li:LIPIcs.APPROX/RANDOM.2025.30,
  author =	{Li, Zhangsong},
  title =	{{Algorithmic Contiguity from Low-Degree Conjecture and Applications in Correlated Random Graphs}},
  booktitle =	{Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2025)},
  pages =	{30:1--30:18},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-397-3},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{353},
  editor =	{Ene, Alina and Chattopadhyay, Eshan},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2025.30},
  URN =		{urn:nbn:de:0030-drops-243965},
  doi =		{10.4230/LIPIcs.APPROX/RANDOM.2025.30},
  annote =	{Keywords: Algorithmic Contiguity, Low-degree Conjecture, Correlated Random Graphs}
}
Document
DOI: 10.4230/LIPIcs.CCC.2025.6
Near-Optimal Averaging Samplers and Matrix Samplers

Authors: Zhiyang Xun and David Zuckerman

Published in: LIPIcs, Volume 339, 40th Computational Complexity Conference (CCC 2025)

Abstract
We present the first efficient averaging sampler that achieves asymptotically optimal randomness complexity and near-optimal sample complexity. For any δ < ε and any constant α > 0, our sampler uses m + O(log (1 / δ)) random bits to output t = O((1/ε² log 1/δ)^{1 + α}) samples Z_1, … , Z_t ∈ {0, 1}^m such that for any function f: {0, 1}^m → [0, 1], Pr[|1/t∑_{i=1}^t f(Z_i) - 𝔼[f]| ≤ ε] ≥ 1 - δ. The randomness complexity is optimal up to a constant factor, and the sample complexity is optimal up to the O((1/(ε²) log 1/(δ))^α) factor. Our technique generalizes to matrix samplers. A matrix sampler is defined similarly, except that f: {0, 1}^m → ℂ^{d×d} and the absolute value is replaced by the spectral norm. Our matrix sampler achieves randomness complexity m + Õ(log(d / δ)) and sample complexity O((1/ε² log d/δ)^{1 + α}) for any constant α > 0, both near-optimal with only a logarithmic factor in randomness complexity and an additional α exponent on the sample complexity. We use known connections with randomness extractors and list-decodable codes to give applications to these objects. Specifically, we give the first extractor construction with optimal seed length up to an arbitrarily small constant factor above 1, when the min-entropy k = β n for a large enough constant β < 1. Finally, we generalize the definition of averaging sampler to any normed vector space.
Cite as

Zhiyang Xun and David Zuckerman. Near-Optimal Averaging Samplers and Matrix Samplers. In 40th Computational Complexity Conference (CCC 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 339, pp. 6:1-6:28, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{xun_et_al:LIPIcs.CCC.2025.6,
  author =	{Xun, Zhiyang and Zuckerman, David},
  title =	{{Near-Optimal Averaging Samplers and Matrix Samplers}},
  booktitle =	{40th Computational Complexity Conference (CCC 2025)},
  pages =	{6:1--6:28},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-379-9},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{339},
  editor =	{Srinivasan, Srikanth},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2025.6},
  URN =		{urn:nbn:de:0030-drops-237001},
  doi =		{10.4230/LIPIcs.CCC.2025.6},
  annote =	{Keywords: Pseudorandomness, Averaging Samplers, Randomness Extractors}
}
Document
Track A: Algorithms, Complexity and Games
DOI: 10.4230/LIPIcs.ICALP.2025.55
Optimal Oblivious Subspace Embeddings with Near-Optimal Sparsity

Authors: Shabarish Chenakkod, Michał Dereziński, and Xiaoyu Dong

Published in: LIPIcs, Volume 334, 52nd International Colloquium on Automata, Languages, and Programming (ICALP 2025)

Abstract
An oblivious subspace embedding is a random m× n matrix Π such that, for any d-dimensional subspace, with high probability Π preserves the norms of all vectors in that subspace within a 1±ε factor. In this work, we give an oblivious subspace embedding with the optimal dimension m = Θ(d/ε²) that has a near-optimal sparsity of Õ(1/ε) non-zero entries per column of Π. This is the first result to nearly match the conjecture of Nelson and Nguyen [FOCS 2013] in terms of the best sparsity attainable by an optimal oblivious subspace embedding, improving on a prior bound of Õ(1/ε⁶) non-zeros per column [Chenakkod et al., STOC 2024]. We further extend our approach to the non-oblivious setting, proposing a new family of Leverage Score Sparsified embeddings with Independent Columns, which yield faster runtimes for matrix approximation and regression tasks. In our analysis, we develop a new method which uses a decoupling argument together with the cumulant method for bounding the edge universality error of isotropic random matrices. To achieve near-optimal sparsity, we combine this general-purpose approach with new trace inequalities that leverage the specific structure of our subspace embedding construction.
Cite as

Shabarish Chenakkod, Michał Dereziński, and Xiaoyu Dong. Optimal Oblivious Subspace Embeddings with Near-Optimal Sparsity. In 52nd International Colloquium on Automata, Languages, and Programming (ICALP 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 334, pp. 55:1-55:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{chenakkod_et_al:LIPIcs.ICALP.2025.55,
  author =	{Chenakkod, Shabarish and Derezi\'{n}ski, Micha{\l} and Dong, Xiaoyu},
  title =	{{Optimal Oblivious Subspace Embeddings with Near-Optimal Sparsity}},
  booktitle =	{52nd International Colloquium on Automata, Languages, and Programming (ICALP 2025)},
  pages =	{55:1--55:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-372-0},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{334},
  editor =	{Censor-Hillel, Keren and Grandoni, Fabrizio and Ouaknine, Jo\"{e}l and Puppis, Gabriele},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2025.55},
  URN =		{urn:nbn:de:0030-drops-234324},
  doi =		{10.4230/LIPIcs.ICALP.2025.55},
  annote =	{Keywords: Randomized linear algebra, matrix sketching, subspace embeddings}
}
Document
Track A: Algorithms, Complexity and Games
DOI: 10.4230/LIPIcs.ICALP.2025.13
Quantum Speedup for Sampling Random Spanning Trees

Authors: Simon Apers, Minbo Gao, Zhengfeng Ji, and Chenghua Liu

Published in: LIPIcs, Volume 334, 52nd International Colloquium on Automata, Languages, and Programming (ICALP 2025)

Abstract
We present a quantum algorithm for sampling random spanning trees from a weighted graph in Õ(√{mn}) time, where n and m denote the number of vertices and edges, respectively. Our algorithm has sublinear runtime for dense graphs and achieves a quantum speedup over the best-known classical algorithm, which runs in Õ(m) time. The approach carefully combines, on one hand, a classical method based on "large-step" random walks for reduced mixing time and, on the other hand, quantum algorithmic techniques, including quantum graph sparsification and a sampling-without-replacement variant of Hamoudi’s multiple-state preparation. We also establish a matching lower bound, proving the optimality of our algorithm up to polylogarithmic factors. These results highlight the potential of quantum computing in accelerating fundamental graph sampling problems.
Cite as

Simon Apers, Minbo Gao, Zhengfeng Ji, and Chenghua Liu. Quantum Speedup for Sampling Random Spanning Trees. In 52nd International Colloquium on Automata, Languages, and Programming (ICALP 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 334, pp. 13:1-13:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{apers_et_al:LIPIcs.ICALP.2025.13,
  author =	{Apers, Simon and Gao, Minbo and Ji, Zhengfeng and Liu, Chenghua},
  title =	{{Quantum Speedup for Sampling Random Spanning Trees}},
  booktitle =	{52nd International Colloquium on Automata, Languages, and Programming (ICALP 2025)},
  pages =	{13:1--13:21},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-372-0},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{334},
  editor =	{Censor-Hillel, Keren and Grandoni, Fabrizio and Ouaknine, Jo\"{e}l and Puppis, Gabriele},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2025.13},
  URN =		{urn:nbn:de:0030-drops-233907},
  doi =		{10.4230/LIPIcs.ICALP.2025.13},
  annote =	{Keywords: Quantum Computing, Quantum Algorithms, Random Spanning Trees}
}
Document
DOI: 10.4230/LIPIcs.ITCS.2025.57
Deterministic Approximation for the Volume of the Truncated Fractional Matching Polytope

Authors: Heng Guo and Vishvajeet N

Published in: LIPIcs, Volume 325, 16th Innovations in Theoretical Computer Science Conference (ITCS 2025)

Abstract
We give a deterministic polynomial-time approximation scheme (FPTAS) for the volume of the truncated fractional matching polytope for graphs of maximum degree Δ, where the truncation is by restricting each variable to the interval [0,(1+δ)/Δ], and δ ≤ C/Δ for some constant C > 0. We also generalise our result to the fractional matching polytope for hypergraphs of maximum degree Δ and maximum hyperedge size k, truncated by [0,(1+δ)/Δ] as well, where δ ≤ CΔ^{-(2k-3)/(k-1)}k^{-1} for some constant C > 0. The latter result generalises both the first result for graphs (when k = 2), and a result by Bencs and Regts (2024) for the truncated independence polytope (when Δ = 2). Our approach is based on the cluster expansion technique.
Cite as

Heng Guo and Vishvajeet N. Deterministic Approximation for the Volume of the Truncated Fractional Matching Polytope. In 16th Innovations in Theoretical Computer Science Conference (ITCS 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 325, pp. 57:1-57:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{guo_et_al:LIPIcs.ITCS.2025.57,
  author =	{Guo, Heng and N, Vishvajeet},
  title =	{{Deterministic Approximation for the Volume of the Truncated Fractional Matching Polytope}},
  booktitle =	{16th Innovations in Theoretical Computer Science Conference (ITCS 2025)},
  pages =	{57:1--57:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-361-4},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{325},
  editor =	{Meka, Raghu},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2025.57},
  URN =		{urn:nbn:de:0030-drops-226858},
  doi =		{10.4230/LIPIcs.ITCS.2025.57},
  annote =	{Keywords: deterministic volume computation, cluster expansion, explicit polytopes}
}
Document
DOI: 10.4230/LIPIcs.APPROX-RANDOM.2015.898
Spectral Norm of Random Kernel Matrices with Applications to Privacy

Authors: Shiva Prasad Kasiviswanathan and Mark Rudelson

Published in: LIPIcs, Volume 40, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2015)

Abstract
Kernel methods are an extremely popular set of techniques used for many important machine learning and data analysis applications. In addition to having good practical performance, these methods are supported by a well-developed theory. Kernel methods use an implicit mapping of the input data into a high dimensional feature space defined by a kernel function, i.e., a function returning the inner product between the images of two data points in the feature space. Central to any kernel method is the kernel matrix, which is built by evaluating the kernel function on a given sample dataset. In this paper, we initiate the study of non-asymptotic spectral properties of random kernel matrices. These are n x n random matrices whose (i,j)th entry is obtained by evaluating the kernel function on x_i and x_j, where x_1,..,x_n are a set of n independent random high-dimensional vectors. Our main contribution is to obtain tight upper bounds on the spectral norm (largest eigenvalue) of random kernel matrices constructed by using common kernel functions such as polynomials and Gaussian radial basis. As an application of these results, we provide lower bounds on the distortion needed for releasing the coefficients of kernel ridge regression under attribute privacy, a general privacy notion which captures a large class of privacy definitions. Kernel ridge regression is standard method for performing non-parametric regression that regularly outperforms traditional regression approaches in various domains. Our privacy distortion lower bounds are the first for any kernel technique, and our analysis assumes realistic scenarios for the input, unlike all previous lower bounds for other release problems which only hold under very restrictive input settings.
Cite as

Shiva Prasad Kasiviswanathan and Mark Rudelson. Spectral Norm of Random Kernel Matrices with Applications to Privacy. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 40, pp. 898-914, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015)

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@InProceedings{kasiviswanathan_et_al:LIPIcs.APPROX-RANDOM.2015.898,
  author =	{Kasiviswanathan, Shiva Prasad and Rudelson, Mark},
  title =	{{Spectral Norm of Random Kernel Matrices with Applications to Privacy}},
  booktitle =	{Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2015)},
  pages =	{898--914},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-939897-89-7},
  ISSN =	{1868-8969},
  year =	{2015},
  volume =	{40},
  editor =	{Garg, Naveen and Jansen, Klaus and Rao, Anup and Rolim, Jos\'{e} D. P.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX-RANDOM.2015.898},
  URN =		{urn:nbn:de:0030-drops-53435},
  doi =		{10.4230/LIPIcs.APPROX-RANDOM.2015.898},
  annote =	{Keywords: Random Kernel Matrices, Spectral Norm, Subguassian Distribution, Data Privacy, Reconstruction Attacks}
}
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