8 Search Results for "Golub, Gene"

Document
DOI: 10.4230/LIPIcs.SoCG.2026.56
Optimal Randomized Clustering of Matrices

Authors: Mustafa Alper Gunes and Assaf Naor

Published in: LIPIcs, Volume 367, 42nd International Symposium on Computational Geometry (SoCG 2026)

Abstract
If X = (𝖬_n(ℝ),‖⋅‖_X) is a unitarily invariant normed space, i.e., ‖𝖴𝖠𝖵‖_X = ‖𝖠‖_X for every matrix 𝖠 ∈ 𝖬_n(ℝ) and every two orthogonal matrices 𝖴,𝖵 ∈ 𝖬_n(ℝ), then we evaluate up to universal constant factors the smallest σ > 0 for which there is a probability distribution over partitions of X into clusters of diameter at most 1 yet for every two matrices 𝖠,𝖡 ∈ 𝖬_n(ℝ) the probability that they fall into distinct clusters is at most σ times the X-distance between 𝖠 and 𝖡. Specifically, we prove that this infimal σ, which is called the separation modulus of X and is denoted SEP(X), satisfies: (1) SEP(X) = Θ(√n⋅ ‖𝖨_n‖_X⋅ diam(B_X)), where 𝖨_n is the n-by-n identity matrix and diam(B_X) is the diameter with respect to the standard Euclidean metric on 𝖬_n(ℝ) of the unit ball B_ X of X. Our proof of (1) proceeds through an asymptotic evaluation of the spectral gap of the Laplacian with Dirichlet boundary conditions on B_ X, which we achieve by exact computations for a Jacobi orthogonal random matrix ensemble. Assuming oracle access to norm evaluations in X, by combining (1) with a new deterministic algorithm for a O(1)-approximation of the diameter of convex bodies in ℝⁿ that are given by a weak membership oracle and are symmetric with respect to coordinate permutations and reflections about the standard axes (this task is famously known to be impossible in the absence of such symmetries), we get an oracle polynomial time algorithm whose output is the separation modulus of X up to universal constant factors. Another example of a consequence of (1) is that for each m ∈ {1,…,n} the separation modulus of the m'th Ky Fan norm on 𝖬_n(ℝ) is bounded from above and from below by universal constant multiples of m√n if m ⩾ √n, and of n if m ⩽ √n. We also deduce from (1) an upper bound on the Lipschitz extension modulus of X that improves over the previously best-known bound even in the special case when X is 𝖬_n(ℝ) equipped with the 𝓁₂ⁿ → 𝓁₂ⁿ operator norm.
Cite as

Mustafa Alper Gunes and Assaf Naor. Optimal Randomized Clustering of Matrices. In 42nd International Symposium on Computational Geometry (SoCG 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 367, pp. 56:1-56:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{gunes_et_al:LIPIcs.SoCG.2026.56,
  author =	{Gunes, Mustafa Alper and Naor, Assaf},
  title =	{{Optimal Randomized Clustering of Matrices}},
  booktitle =	{42nd International Symposium on Computational Geometry (SoCG 2026)},
  pages =	{56:1--56:18},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-418-5},
  ISSN =	{1868-8969},
  year =	{2026},
  volume =	{367},
  editor =	{Ahn, Hee-Kap and Hoffmann, Michael and Nayyeri, Amir},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2026.56},
  URN =		{urn:nbn:de:0030-drops-258624},
  doi =		{10.4230/LIPIcs.SoCG.2026.56},
  annote =	{Keywords: Clustering, Unitarily Invariant Matrix Norms, Oracle Polynomial Time Approximation Algorithms for Radii of Convex Bodies, Extension of Lipschitz Functions, Random Matrices, Spectrum of the Laplacian with Dirichlet Boundary Conditions, Reverse Isoperimetry}
}
Document
DOI: 10.4230/LIPIcs.FUN.2026.14
The Careless Coupon Collector’s Problem

Authors: Emilio Cruciani and Aditi Dudeja

Published in: LIPIcs, Volume 366, 13th International Conference on Fun with Algorithms (FUN 2026)

Abstract
We initiate the study of the Careless Coupon Collector’s Problem (CCCP), a novel variation of the classical coupon collector, that we envision as a model for information systems such as web crawlers, dynamic caches, and fault-resilient networks. In CCCP, a collector attempts to gather n distinct coupon types by obtaining one coupon type uniformly at random in each discrete round, however the collector is careless: at the end of each round, each collected coupon type is independently lost with probability p. We analyze the number of rounds required to complete the collection as a function of n and p. In particular, we show that it transitions from Θ(n ln n) when p = o((ln n)/n²) up to Θ(((np)/(1-p))ⁿ) when p = ω(1/n) in multiple distinct phases. Interestingly, when p = c/n, the process remains in a metastable phase, where the fraction of collected coupon types is concentrated around 1/(1+c) with probability 1-o(1), for a time window of length e^{Θ(n)}. Finally, we give an algorithm that computes the expected completion time of CCCP in O(n²) time.
Cite as

Emilio Cruciani and Aditi Dudeja. The Careless Coupon Collector’s Problem. In 13th International Conference on Fun with Algorithms (FUN 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 366, pp. 14:1-14:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{cruciani_et_al:LIPIcs.FUN.2026.14,
  author =	{Cruciani, Emilio and Dudeja, Aditi},
  title =	{{The Careless Coupon Collector’s Problem}},
  booktitle =	{13th International Conference on Fun with Algorithms (FUN 2026)},
  pages =	{14:1--14:21},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-417-8},
  ISSN =	{1868-8969},
  year =	{2026},
  volume =	{366},
  editor =	{Iacono, John},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FUN.2026.14},
  URN =		{urn:nbn:de:0030-drops-257333},
  doi =		{10.4230/LIPIcs.FUN.2026.14},
  annote =	{Keywords: Coupon Collector, Markov Chains, Metastability}
}
Document
Brief Announcement
DOI: 10.4230/LIPIcs.DISC.2025.60
Brief Announcement: Distributed Sparsest Cut via Eigenvalue Estimation

Authors: Yannic Maus and Tijn de Vos

Published in: LIPIcs, Volume 356, 39th International Symposium on Distributed Computing (DISC 2025)

Abstract
We give new, improved bounds for approximating the sparsest cut value or in other words the conductance ϕ of a graph in the CONGEST model. As our main result, we present an algorithm running in O(log² n/ϕ) rounds in which every vertex outputs a value ̃ ϕ satisfying ϕ ≤ ̃ ϕ ≤ √{2.01ϕ}. In most regimes, our algorithm improves significantly over the previously fastest algorithm for the problem [Chen, Meierhans, Probst Gutenberg, Saranurak; SODA 25]. Additionally, our result generalizes to k-way conductance. We obtain these results, by approximating the eigenvalues of the normalized Laplacian matrix L: = I-Deg^{-1/2}ADeg^ {-1/2}, where, A is the adjacency matrix and Deg is the diagonal matrix with the weighted degrees on the diagonal. We show our algorithms are near-optimal by proving a lower bound for computing the smallest non-trivial eigenvalue of L, even in the stronger LOCAL model The previous state of the art sparsest cut algorithm is in the technical realm of expander decompositions. Our algorithms, on the other hand, are relatively simple and easy to implement. At the core, they rely on the well-known power method, which comes down to repeatedly multiplying the Laplacian with a vector. This operation can be performed in a single round in the CONGEST model. All our algorithms apply to weighted, undirected graphs. Our lower bounds apply even in unweighted graphs.
Cite as

Yannic Maus and Tijn de Vos. Brief Announcement: Distributed Sparsest Cut via Eigenvalue Estimation. In 39th International Symposium on Distributed Computing (DISC 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 356, pp. 60:1-60:7, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{maus_et_al:LIPIcs.DISC.2025.60,
  author =	{Maus, Yannic and de Vos, Tijn},
  title =	{{Brief Announcement: Distributed Sparsest Cut via Eigenvalue Estimation}},
  booktitle =	{39th International Symposium on Distributed Computing (DISC 2025)},
  pages =	{60:1--60:7},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-402-4},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{356},
  editor =	{Kowalski, Dariusz R.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.DISC.2025.60},
  URN =		{urn:nbn:de:0030-drops-248763},
  doi =		{10.4230/LIPIcs.DISC.2025.60},
  annote =	{Keywords: CONGEST, Sparsest Cut, Laplacian, Eigenvalues, Spectral Graph Theory}
}
Document
Track A: Algorithms, Complexity and Games
DOI: 10.4230/LIPIcs.ICALP.2025.131
Deterministic Complexity Analysis of Hermitian Eigenproblems

Authors: Aleksandros Sobczyk

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

Abstract
In this work we revisit the arithmetic and bit complexity of Hermitian eigenproblems. Recently, [BGVKS, FOCS 2020] proved that a (non-Hermitian) matrix A can be diagonalized with a randomized algorithm in O(n^{ω}log²(n/ε)) arithmetic operations, where ω≲ 2.371 is the square matrix multiplication exponent, and [Shah, SODA 2025] significantly improved the bit complexity for the Hermitian case. Our main goal is to obtain similar deterministic complexity bounds for various Hermitian eigenproblems. In the Real RAM model, we show that a Hermitian matrix can be diagonalized deterministically in O(n^{ω}log(n)+n²polylog(n/ε)) arithmetic operations, improving the classic deterministic Õ(n³) algorithms, and derandomizing the aforementioned state-of-the-art. The main technical step is a complete, detailed analysis of a well-known divide-and-conquer tridiagonal eigensolver of Gu and Eisenstat [GE95], when accelerated with the Fast Multipole Method, asserting that it can accurately diagonalize a symmetric tridiagonal matrix in nearly-O(n²) operations. In finite precision, we show that an algorithm by Schönhage [Sch72] to reduce a Hermitian matrix to tridiagonal form is stable in the floating point model, using O(log(n/ε)) bits of precision. This leads to a deterministic algorithm to compute all the eigenvalues of a Hermitian matrix in O(n^{ω}ℱ(log(n/ε)) + n²polylog(n/ε)) bit operations, where ℱ(b) ∈ Õ(b) is the bit complexity of a single floating point operation on b bits. This improves the best known Õ(n³) deterministic and O(n^{ω}log²(n/ε)ℱ(log(n/ε))) randomized complexities. We conclude with some other useful subroutines such as computing spectral gaps, condition numbers, and spectral projectors, and with some open problems.
Cite as

Aleksandros Sobczyk. Deterministic Complexity Analysis of Hermitian Eigenproblems. In 52nd International Colloquium on Automata, Languages, and Programming (ICALP 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 334, pp. 131:1-131:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{sobczyk:LIPIcs.ICALP.2025.131,
  author =	{Sobczyk, Aleksandros},
  title =	{{Deterministic Complexity Analysis of Hermitian Eigenproblems}},
  booktitle =	{52nd International Colloquium on Automata, Languages, and Programming (ICALP 2025)},
  pages =	{131:1--131:21},
  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.2025.131},
  URN =		{urn:nbn:de:0030-drops-235081},
  doi =		{10.4230/LIPIcs.ICALP.2025.131},
  annote =	{Keywords: Hermitian eigenproblem, eigenvalues, SVD, tridiagonal reduction, matrix multiplication time, diagonalization, bit complexity}
}
Document
Track A: Algorithms, Complexity and Games
DOI: 10.4230/LIPIcs.ICALP.2025.6
Decremental (1+ε)-Approximate Maximum Eigenvector: Dynamic Power Method

Authors: Deeksha Adil and Thatchaphol Saranurak

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

Abstract
We present a dynamic algorithm for maintaining (1+ε)-approximate maximum eigenvector and eigenvalue of a positive semi-definite matrix A undergoing decreasing updates, i.e., updates which may only decrease eigenvalues. Given a vector v updating A ← A-vv^⊤, our algorithm takes Õ(nnz(v)) amortized update time, i.e., polylogarithmic per non-zeros in the update vector. Our technique is based on a novel analysis of the influential power method in the dynamic setting. The two previous sets of techniques have the following drawbacks (1) algebraic techniques can maintain exact solutions but their update time is at least polynomial per non-zeros, and (2) sketching techniques admit polylogarithmic update time but suffer from a crude additive approximation. Our algorithm exploits an oblivious adversary. Interestingly, we show that any algorithm with polylogarithmic update time per non-zeros that works against an adaptive adversary and satisfies an additional natural property would imply a breakthrough for checking psd-ness of matrices in Õ(n²) time, instead of O(n^ω) time.
Cite as

Deeksha Adil and Thatchaphol Saranurak. Decremental (1+ε)-Approximate Maximum Eigenvector: Dynamic Power Method. In 52nd International Colloquium on Automata, Languages, and Programming (ICALP 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 334, pp. 6:1-6:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{adil_et_al:LIPIcs.ICALP.2025.6,
  author =	{Adil, Deeksha and Saranurak, Thatchaphol},
  title =	{{Decremental (1+\epsilon)-Approximate Maximum Eigenvector: Dynamic Power Method}},
  booktitle =	{52nd International Colloquium on Automata, Languages, and Programming (ICALP 2025)},
  pages =	{6:1--6:19},
  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.2025.6},
  URN =		{urn:nbn:de:0030-drops-233834},
  doi =		{10.4230/LIPIcs.ICALP.2025.6},
  annote =	{Keywords: Power Method, Dynamic Algorithms}
}
Document
Track A: Algorithms, Complexity and Games
DOI: 10.4230/LIPIcs.ICALP.2025.29
Guessing Efficiently for Constrained Subspace Approximation

Authors: Aditya Bhaskara, Sepideh Mahabadi, Madhusudhan Reddy Pittu, Ali Vakilian, and David P. Woodruff

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

Abstract
In this paper we study constrained subspace approximation problem. Given a set of n points {a₁,…,a_n} in ℝ^d, the goal of the subspace approximation problem is to find a k dimensional subspace that best approximates the input points. More precisely, for a given p ≥ 1, we aim to minimize the pth power of the 𝓁_p norm of the error vector (‖a₁-Pa₁‖,…,‖a_n-Pa_n‖), where P denotes the projection matrix onto the subspace and the norms are Euclidean. In constrained subspace approximation (CSA), we additionally have constraints on the projection matrix P. In its most general form, we require P to belong to a given subset 𝒮 that is described explicitly or implicitly. We introduce a general framework for constrained subspace approximation. Our approach, that we term coreset-guess-solve, yields either (1+ε)-multiplicative or ε-additive approximations for a variety of constraints. We show that it provides new algorithms for partition-constrained subspace approximation with applications to fair subspace approximation, k-means clustering, and projected non-negative matrix factorization, among others. Specifically, while we reconstruct the best known bounds for k-means clustering in Euclidean spaces, we improve the known results for the remainder of the problems.
Cite as

Aditya Bhaskara, Sepideh Mahabadi, Madhusudhan Reddy Pittu, Ali Vakilian, and David P. Woodruff. Guessing Efficiently for Constrained Subspace Approximation. In 52nd International Colloquium on Automata, Languages, and Programming (ICALP 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 334, pp. 29:1-29:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{bhaskara_et_al:LIPIcs.ICALP.2025.29,
  author =	{Bhaskara, Aditya and Mahabadi, Sepideh and Pittu, Madhusudhan Reddy and Vakilian, Ali and Woodruff, David P.},
  title =	{{Guessing Efficiently for Constrained Subspace Approximation}},
  booktitle =	{52nd International Colloquium on Automata, Languages, and Programming (ICALP 2025)},
  pages =	{29:1--29:20},
  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.2025.29},
  URN =		{urn:nbn:de:0030-drops-234068},
  doi =		{10.4230/LIPIcs.ICALP.2025.29},
  annote =	{Keywords: parameterized complexity, low rank approximation, fairness, non-negative matrix factorization, clustering}
}
Document
Track B: Automata, Logic, Semantics, and Theory of Programming
DOI: 10.4230/LIPIcs.ICALP.2025.148
Holant* Dichotomy on Domain Size 3: A Geometric Perspective

Authors: Jin-Yi Cai and Jin Soo Ihm

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

Abstract
Holant problems are a general framework to study the computational complexity of counting problems. It is a more expressive framework than counting constraint satisfaction problems (CSP) which are in turn more expressive than counting graph homomorphisms (GH). In this paper, we prove the first complexity dichotomy of Holant^*₃(ℱ) where ℱ is an arbitrary set of symmetric, real valued constraint functions on domain size 3. We give an explicit tractability criterion and prove that, if ℱ satisfies this criterion then Holant^*₃(ℱ) is polynomial time computable, and otherwise it is #P-hard, with no intermediate cases. We show that the geometry of the tensor decomposition of the constraint functions plays a central role in the formulation as well as the structural internal logic of the dichotomy.
Cite as

Jin-Yi Cai and Jin Soo Ihm. Holant* Dichotomy on Domain Size 3: A Geometric Perspective. In 52nd International Colloquium on Automata, Languages, and Programming (ICALP 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 334, pp. 148:1-148:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{cai_et_al:LIPIcs.ICALP.2025.148,
  author =	{Cai, Jin-Yi and Ihm, Jin Soo},
  title =	{{Holant* Dichotomy on Domain Size 3: A Geometric Perspective}},
  booktitle =	{52nd International Colloquium on Automata, Languages, and Programming (ICALP 2025)},
  pages =	{148:1--148:18},
  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.2025.148},
  URN =		{urn:nbn:de:0030-drops-235254},
  doi =		{10.4230/LIPIcs.ICALP.2025.148},
  annote =	{Keywords: Holant problem, Complexity dichotomy, Higher domain}
}
Document
DOI: 10.4230/DagSemProc.07071.17
Three results on the PageRank vector: eigenstructure, sensitivity, and the derivative

Authors: David Gleich, Peter Glynn, Gene Golub, and Chen Greif

Published in: Dagstuhl Seminar Proceedings, Volume 7071, Web Information Retrieval and Linear Algebra Algorithms (2007)

Abstract
The three results on the PageRank vector are preliminary but shed light on the eigenstructure of a PageRank modified Markov chain and what happens when changing the teleportation parameter in the PageRank model. Computations with the derivative of the PageRank vector with respect to the teleportation parameter show predictive ability and identify an interesting set of pages from Wikipedia.
Cite as

David Gleich, Peter Glynn, Gene Golub, and Chen Greif. Three results on the PageRank vector: eigenstructure, sensitivity, and the derivative. In Web Information Retrieval and Linear Algebra Algorithms. Dagstuhl Seminar Proceedings, Volume 7071, pp. 1-10, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2007)

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@InProceedings{gleich_et_al:DagSemProc.07071.17,
  author =	{Gleich, David and Glynn, Peter and Golub, Gene and Greif, Chen},
  title =	{{Three results on the PageRank vector:  eigenstructure, sensitivity, and the derivative}},
  booktitle =	{Web Information Retrieval and Linear Algebra Algorithms},
  pages =	{1--10},
  series =	{Dagstuhl Seminar Proceedings (DagSemProc)},
  ISSN =	{1862-4405},
  year =	{2007},
  volume =	{7071},
  editor =	{Andreas Frommer and Michael W. Mahoney and Daniel B. Szyld},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/DagSemProc.07071.17},
  URN =		{urn:nbn:de:0030-drops-10615},
  doi =		{10.4230/DagSemProc.07071.17},
  annote =	{Keywords: PageRank, PageRank derivative, PageRank sensitivity, PageRank eigenstructure}
}
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