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**Published in:** LIPIcs, Volume 287, 15th Innovations in Theoretical Computer Science Conference (ITCS 2024)

Let S ∈ ℝ^{n × n} be any matrix satisfying ‖1-S‖₂ ≤ εn, where 1 is the all ones matrix and ‖⋅‖₂ is the spectral norm. It is well-known that there exists S with just O(n/ε²) non-zero entries achieving this guarantee: we can let 𝐒 be the scaled adjacency matrix of a Ramanujan expander graph. We show that, beyond giving a sparse approximation to the all ones matrix, 𝐒 yields a universal sparsifier for any positive semidefinite (PSD) matrix. In particular, for any PSD A ∈ ℝ^{n×n} which is normalized so that its entries are bounded in magnitude by 1, we show that ‖A-A∘S‖₂ ≤ ε n, where ∘ denotes the entrywise (Hadamard) product. Our techniques also yield universal sparsifiers for non-PSD matrices. In this case, we show that if S satisfies ‖1-S‖₂ ≤ (ε²n)/(c log²(1/ε)) for some sufficiently large constant c, then ‖A-A∘S‖₂ ≤ ε⋅max(n,‖ A‖₁), where ‖A‖₁ is the nuclear norm. Again letting 𝐒 be a scaled Ramanujan graph adjacency matrix, this yields a sparsifier with Õ(n/ε⁴) entries. We prove that the above universal sparsification bounds for both PSD and non-PSD matrices are tight up to logarithmic factors.
Since 𝐀∘𝐒 can be constructed deterministically without reading all of A, our result for PSD matrices derandomizes and improves upon established results for randomized matrix sparsification, which require sampling a random subset of O((n log n)/ε²) entries and only give an approximation to any fixed A with high probability. We further show that any randomized algorithm must read at least Ω(n/ε²) entries to spectrally approximate general A to error εn, thus proving that these existing randomized algorithms are optimal up to logarithmic factors. We leverage our deterministic sparsification results to give the first {deterministic algorithms} for several problems, including singular value and singular vector approximation and positive semidefiniteness testing, that run in faster than matrix multiplication time. This partially addresses a significant gap between randomized and deterministic algorithms for fast linear algebraic computation.
Finally, if A ∈ {-1,0,1}^{n × n} is PSD, we show that a spectral approximation Ã with ‖A-Ã‖₂ ≤ ε n can be obtained by deterministically reading Õ(n/ε) entries of A. This improves the 1/ε dependence on our result for general PSD matrices by a quadratic factor and is information-theoretically optimal up to a logarithmic factor.

Rajarshi Bhattacharjee, Gregory Dexter, Cameron Musco, Archan Ray, Sushant Sachdeva, and David P. Woodruff. Universal Matrix Sparsifiers and Fast Deterministic Algorithms for Linear Algebra. In 15th Innovations in Theoretical Computer Science Conference (ITCS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 287, pp. 13:1-13:24, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{bhattacharjee_et_al:LIPIcs.ITCS.2024.13, author = {Bhattacharjee, Rajarshi and Dexter, Gregory and Musco, Cameron and Ray, Archan and Sachdeva, Sushant and Woodruff, David P.}, title = {{Universal Matrix Sparsifiers and Fast Deterministic Algorithms for Linear Algebra}}, booktitle = {15th Innovations in Theoretical Computer Science Conference (ITCS 2024)}, pages = {13:1--13:24}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-309-6}, ISSN = {1868-8969}, year = {2024}, volume = {287}, editor = {Guruswami, Venkatesan}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2024.13}, URN = {urn:nbn:de:0030-drops-195415}, doi = {10.4230/LIPIcs.ITCS.2024.13}, annote = {Keywords: sublinear algorithms, randomized linear algebra, spectral sparsification, expanders} }

Document

**Published in:** LIPIcs, Volume 287, 15th Innovations in Theoretical Computer Science Conference (ITCS 2024)

Oblivious routing is a well-studied paradigm that uses static precomputed routing tables for selecting routing paths within a network. Existing oblivious routing schemes with polylogarithmic competitive ratio for general networks are tree-based, in the sense that routing is performed according to a convex combination of trees. However, this restriction to trees leads to a construction that has time quadratic in the size of the network and does not parallelize well.
In this paper we study oblivious routing schemes based on electrical routing. In particular, we show that general networks with n vertices and m edges admit a routing scheme that has competitive ratio O(log² n) and consists of a convex combination of only O(√m) electrical routings. This immediately leads to an improved construction algorithm with time Õ(m^{3/2}) that can also be implemented in parallel with Õ(√m) depth.

Gramoz Goranci, Monika Henzinger, Harald Räcke, Sushant Sachdeva, and A. R. Sricharan. Electrical Flows for Polylogarithmic Competitive Oblivious Routing. In 15th Innovations in Theoretical Computer Science Conference (ITCS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 287, pp. 55:1-55:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{goranci_et_al:LIPIcs.ITCS.2024.55, author = {Goranci, Gramoz and Henzinger, Monika and R\"{a}cke, Harald and Sachdeva, Sushant and Sricharan, A. R.}, title = {{Electrical Flows for Polylogarithmic Competitive Oblivious Routing}}, booktitle = {15th Innovations in Theoretical Computer Science Conference (ITCS 2024)}, pages = {55:1--55:22}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-309-6}, ISSN = {1868-8969}, year = {2024}, volume = {287}, editor = {Guruswami, Venkatesan}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2024.55}, URN = {urn:nbn:de:0030-drops-195830}, doi = {10.4230/LIPIcs.ITCS.2024.55}, annote = {Keywords: oblivious routing, electrical flows} }

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Track A: Algorithms, Complexity and Games

**Published in:** LIPIcs, Volume 198, 48th International Colloquium on Automata, Languages, and Programming (ICALP 2021)

We give almost-linear-time algorithms for constructing sparsifiers with n poly(log n) edges that approximately preserve weighted (𝓁²₂ + 𝓁^p_p) flow or voltage objectives on graphs. For flow objectives, this is the first sparsifier construction for such mixed objectives beyond unit 𝓁_p weights, and is based on expander decompositions. For voltage objectives, we give the first sparsifier construction for these objectives, which we build using graph spanners and leverage score sampling. Together with the iterative refinement framework of [Adil et al, SODA 2019], and a new multiplicative-weights based constant-approximation algorithm for mixed-objective flows or voltages, we show how to find (1+2^{-poly(log n)}) approximations for weighted 𝓁_p-norm minimizing flows or voltages in p(m^{1+o(1)} + n^{4/3 + o(1)}) time for p = ω(1), which is almost-linear for graphs that are slightly dense (m ≥ n^{4/3 + o(1)}).

Deeksha Adil, Brian Bullins, Rasmus Kyng, and Sushant Sachdeva. Almost-Linear-Time Weighted 𝓁_p-Norm Solvers in Slightly Dense Graphs via Sparsification. In 48th International Colloquium on Automata, Languages, and Programming (ICALP 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 198, pp. 9:1-9:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{adil_et_al:LIPIcs.ICALP.2021.9, author = {Adil, Deeksha and Bullins, Brian and Kyng, Rasmus and Sachdeva, Sushant}, title = {{Almost-Linear-Time Weighted 𝓁\underlinep-Norm Solvers in Slightly Dense Graphs via Sparsification}}, booktitle = {48th International Colloquium on Automata, Languages, and Programming (ICALP 2021)}, pages = {9:1--9:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-195-5}, ISSN = {1868-8969}, year = {2021}, volume = {198}, editor = {Bansal, Nikhil and Merelli, Emanuela and Worrell, James}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2021.9}, URN = {urn:nbn:de:0030-drops-140782}, doi = {10.4230/LIPIcs.ICALP.2021.9}, annote = {Keywords: Weighted 𝓁\underlinep-norm, Sparsification, Spanners, Iterative Refinement} }

Document

**Published in:** LIPIcs, Volume 94, 9th Innovations in Theoretical Computer Science Conference (ITCS 2018)

We study whether a depth two neural network can learn another
depth two network using gradient descent. Assuming a linear output node, we show that the question of whether gradient descent converges to the target function is equivalent to the following question in electrodynamics: Given k fixed protons in R^d, and k electrons, each moving due to the attractive force from the protons and repulsive force from the remaining electrons, whether at equilibrium all the electrons will be matched up with the protons, up to a permutation. Under the standard electrical force, this follows from the classic Earnshaw's theorem. In our setting,
the force is determined by the activation function and the
input distribution. Building on this equivalence, we prove the
existence of an activation function such that gradient descent learns at least one of the hidden nodes in the target network.
Iterating, we show that gradient descent can be used to learn the entire network one node at a time.

Rina Panigrahy, Ali Rahimi, Sushant Sachdeva, and Qiuyi Zhang. Convergence Results for Neural Networks via Electrodynamics. In 9th Innovations in Theoretical Computer Science Conference (ITCS 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 94, pp. 22:1-22:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{panigrahy_et_al:LIPIcs.ITCS.2018.22, author = {Panigrahy, Rina and Rahimi, Ali and Sachdeva, Sushant and Zhang, Qiuyi}, title = {{Convergence Results for Neural Networks via Electrodynamics}}, booktitle = {9th Innovations in Theoretical Computer Science Conference (ITCS 2018)}, pages = {22:1--22:19}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-060-6}, ISSN = {1868-8969}, year = {2018}, volume = {94}, editor = {Karlin, Anna R.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2018.22}, URN = {urn:nbn:de:0030-drops-83521}, doi = {10.4230/LIPIcs.ITCS.2018.22}, annote = {Keywords: Deep Learning, Learning Theory, Non-convex Optimization} }

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