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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} }

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Track A: Algorithms, Complexity and Games

**Published in:** LIPIcs, Volume 261, 50th International Colloquium on Automata, Languages, and Programming (ICALP 2023)

We study the problem of approximating the eigenspectrum of a symmetric matrix A ∈ ℝ^{n×n} with bounded entries (i.e., ‖A‖_∞ ≤ 1). We present a simple sublinear time algorithm that approximates all eigenvalues of A up to additive error ±εn using those of a randomly sampled Õ((log³ n)/ε³)×Õ((log³ n)/ε³) principal submatrix. Our result can be viewed as a concentration bound on the complete eigenspectrum of a random submatrix, significantly extending known bounds on just the singular values (the magnitudes of the eigenvalues). We give improved error bounds of ± ε √{nnz(A)} and ±ε‖A‖_F when the rows of A can be sampled with probabilities proportional to their sparsities or their squared 𝓁₂ norms respectively. Here nnz(A) is the number of non-zero entries in A and ‖A‖_F is its Frobenius norm. Even for the strictly easier problems of approximating the singular values or testing the existence of large negative eigenvalues (Bakshi, Chepurko, and Jayaram, FOCS '20), our results are the first that take advantage of non-uniform sampling to give improved error bounds. From a technical perspective, our results require several new eigenvalue concentration and perturbation bounds for matrices with bounded entries. Our non-uniform sampling bounds require a new algorithmic approach, which judiciously zeroes out entries of a randomly sampled submatrix to reduce variance, before computing the eigenvalues of that submatrix as estimates for those of A. We complement our theoretical results with numerical simulations, which demonstrate the effectiveness of our algorithms in practice.

Rajarshi Bhattacharjee, Gregory Dexter, Petros Drineas, Cameron Musco, and Archan Ray. Sublinear Time Eigenvalue Approximation via Random Sampling. In 50th International Colloquium on Automata, Languages, and Programming (ICALP 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 261, pp. 21:1-21:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{bhattacharjee_et_al:LIPIcs.ICALP.2023.21, author = {Bhattacharjee, Rajarshi and Dexter, Gregory and Drineas, Petros and Musco, Cameron and Ray, Archan}, title = {{Sublinear Time Eigenvalue Approximation via Random Sampling}}, booktitle = {50th International Colloquium on Automata, Languages, and Programming (ICALP 2023)}, pages = {21:1--21:18}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-278-5}, ISSN = {1868-8969}, year = {2023}, volume = {261}, editor = {Etessami, Kousha and Feige, Uriel 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.2023.21}, URN = {urn:nbn:de:0030-drops-180733}, doi = {10.4230/LIPIcs.ICALP.2023.21}, annote = {Keywords: sublinear algorithms, eigenvalue approximation, randomized linear algebra} }

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**Published in:** LIPIcs, Volume 244, 30th Annual European Symposium on Algorithms (ESA 2022)

We study the problem of estimating the number of edges in an n-vertex graph, accessed via the Bipartite Independent Set query model introduced by Beame et al. (TALG '20). In this model, each query returns a Boolean, indicating the existence of at least one edge between two specified sets of nodes. We present a non-adaptive algorithm that returns a (1± ε) relative error approximation to the number of edges, with query complexity Õ(ε^{-5}log⁵ n), where Õ(⋅) hides poly(log log n) dependencies. This is the first non-adaptive algorithm in this setting achieving poly(1/ε,log n) query complexity. Prior work requires Ω(log² n) rounds of adaptivity. We avoid this by taking a fundamentally different approach, inspired by work on single-pass streaming algorithms. Moreover, for constant ε, our query complexity significantly improves on the best known adaptive algorithm due to Bhattacharya et al. (STACS '22), which requires O(ε^{-2} log^{11} n) queries. Building on our edge estimation result, we give the first {non-adaptive} algorithm for outputting a nearly uniformly sampled edge with query complexity Õ(ε^{-6} log⁶ n), improving on the works of Dell et al. (SODA '20) and Bhattacharya et al. (STACS '22), which require Ω(log³ n) rounds of adaptivity. Finally, as a consequence of our edge sampling algorithm, we obtain a Õ(n log^8 n) query algorithm for connectivity, using two rounds of adaptivity. This improves on a three-round algorithm of Assadi et al. (ESA '21) and is tight; there is no non-adaptive algorithm for connectivity making o(n²) queries.

Raghavendra Addanki, Andrew McGregor, and Cameron Musco. Non-Adaptive Edge Counting and Sampling via Bipartite Independent Set Queries. In 30th Annual European Symposium on Algorithms (ESA 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 244, pp. 2:1-2:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{addanki_et_al:LIPIcs.ESA.2022.2, author = {Addanki, Raghavendra and McGregor, Andrew and Musco, Cameron}, title = {{Non-Adaptive Edge Counting and Sampling via Bipartite Independent Set Queries}}, booktitle = {30th Annual European Symposium on Algorithms (ESA 2022)}, pages = {2:1--2:16}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-247-1}, ISSN = {1868-8969}, year = {2022}, volume = {244}, editor = {Chechik, Shiri and Navarro, Gonzalo and Rotenberg, Eva and Herman, Grzegorz}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2022.2}, URN = {urn:nbn:de:0030-drops-169400}, doi = {10.4230/LIPIcs.ESA.2022.2}, annote = {Keywords: sublinear graph algorithms, bipartite independent set queries, edge sampling and counting, graph connectivity, query adaptivity} }

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**Published in:** LIPIcs, Volume 185, 12th Innovations in Theoretical Computer Science Conference (ITCS 2021)

In the masked low-rank approximation problem, one is given data matrix A ∈ ℝ^{n × n} and binary mask matrix W ∈ {0,1}^{n × n}. The goal is to find a rank-k matrix L for which:
cost(L) := ∑_{i=1}^n ∑_{j=1}^n W_{i,j} ⋅ (A_{i,j} - L_{i,j})² ≤ OPT + ε ‖A‖_F²,
where OPT = min_{rank-k L̂} cost(L̂) and ε is a given error parameter. Depending on the choice of W, the above problem captures factor analysis, low-rank plus diagonal decomposition, robust PCA, low-rank matrix completion, low-rank plus block matrix approximation, low-rank recovery from monotone missing data, and a number of other important problems. Many of these problems are NP-hard, and while algorithms with provable guarantees are known in some cases, they either 1) run in time n^Ω(k²/ε) or 2) make strong assumptions, for example, that A is incoherent or that the entries in W are chosen independently and uniformly at random.
In this work, we show that a common polynomial time heuristic, which simply sets A to 0 where W is 0, and then finds a standard low-rank approximation, yields bicriteria approximation guarantees for this problem. In particular, for rank k' > k depending on the public coin partition number of W, the heuristic outputs rank-k' L with cost(L) ≤ OPT + ε ‖A‖_F². This partition number is in turn bounded by the randomized communication complexity of W, when interpreted as a two-player communication matrix. For many important cases, including all those listed above, this yields bicriteria approximation guarantees with rank k' = k ⋅ poly(log n/ε).
Beyond this result, we show that different notions of communication complexity yield bicriteria algorithms for natural variants of masked low-rank approximation. For example, multi-player number-in-hand communication complexity connects to masked tensor decomposition and non-deterministic communication complexity to masked Boolean low-rank factorization.

Cameron Musco, Christopher Musco, and David P. Woodruff. Simple Heuristics Yield Provable Algorithms for Masked Low-Rank Approximation. In 12th Innovations in Theoretical Computer Science Conference (ITCS 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 185, pp. 6:1-6:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{musco_et_al:LIPIcs.ITCS.2021.6, author = {Musco, Cameron and Musco, Christopher and Woodruff, David P.}, title = {{Simple Heuristics Yield Provable Algorithms for Masked Low-Rank Approximation}}, booktitle = {12th Innovations in Theoretical Computer Science Conference (ITCS 2021)}, pages = {6:1--6:20}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-177-1}, ISSN = {1868-8969}, year = {2021}, volume = {185}, editor = {Lee, James R.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2021.6}, URN = {urn:nbn:de:0030-drops-135452}, doi = {10.4230/LIPIcs.ITCS.2021.6}, annote = {Keywords: low-rank approximation, communication complexity, weighted low-rank approximation, bicriteria approximation algorithms} }

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**Published in:** LIPIcs, Volume 179, 34th International Symposium on Distributed Computing (DISC 2020)

We initiate the study of biologically-inspired spiking neural networks from the perspective of streaming algorithms. Like computers, human brains face memory limitations, which pose a significant obstacle when processing large scale and dynamically changing data. In computer science, these challenges are captured by the well-known streaming model, which can be traced back to Munro and Paterson `78 and has had significant impact in theory and beyond. In the classical streaming setting, one must compute a function f of a stream of updates 𝒮 = {u₁,…,u_m}, given restricted single-pass access to the stream. The primary complexity measure is the space used by the algorithm.
In contrast to the large body of work on streaming algorithms, relatively little is known about the computational aspects of data processing in spiking neural networks. In this work, we seek to connect these two models, leveraging techniques developed for streaming algorithms to better understand neural computation. Our primary goal is to design networks for various computational tasks using as few auxiliary (non-input or output) neurons as possible. The number of auxiliary neurons can be thought of as the "space" required by the network.
Previous algorithmic work in spiking neural networks has many similarities with streaming algorithms. However, the connection between these two space-limited models has not been formally addressed. We take the first steps towards understanding this connection. On the upper bound side, we design neural algorithms based on known streaming algorithms for fundamental tasks, including distinct elements, approximate median, and heavy hitters. The number of neurons in our solutions almost match the space bounds of the corresponding streaming algorithms. As a general algorithmic primitive, we show how to implement the important streaming technique of linear sketching efficiently in spiking neural networks. On the lower bound side, we give a generic reduction, showing that any space-efficient spiking neural network can be simulated by a space-efficient streaming algorithm. This reduction lets us translate streaming-space lower bounds into nearly matching neural-space lower bounds, establishing a close connection between the two models.

Yael Hitron, Cameron Musco, and Merav Parter. Spiking Neural Networks Through the Lens of Streaming Algorithms. In 34th International Symposium on Distributed Computing (DISC 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 179, pp. 10:1-10:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{hitron_et_al:LIPIcs.DISC.2020.10, author = {Hitron, Yael and Musco, Cameron and Parter, Merav}, title = {{Spiking Neural Networks Through the Lens of Streaming Algorithms}}, booktitle = {34th International Symposium on Distributed Computing (DISC 2020)}, pages = {10:1--10:18}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-168-9}, ISSN = {1868-8969}, year = {2020}, volume = {179}, editor = {Attiya, Hagit}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.DISC.2020.10}, URN = {urn:nbn:de:0030-drops-130882}, doi = {10.4230/LIPIcs.DISC.2020.10}, annote = {Keywords: Biological distributed algorithms, Spiking neural networks, Streaming algorithms} }

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**Published in:** LIPIcs, Volume 151, 11th Innovations in Theoretical Computer Science Conference (ITCS 2020)

We study input compression in a biologically inspired model of neural computation. We demonstrate that a network consisting of a random projection step (implemented via random synaptic connectivity) followed by a sparsification step (implemented via winner-take-all competition) can reduce well-separated high-dimensional input vectors to well-separated low-dimensional vectors. By augmenting our network with a third module, we can efficiently map each input (along with any small perturbations of the input) to a unique representative neuron, solving a neural clustering problem.
Both the size of our network and its processing time, i.e., the time it takes the network to compute the compressed output given a presented input, are independent of the (potentially large) dimension of the input patterns and depend only on the number of distinct inputs that the network must encode and the pairwise relative Hamming distance between these inputs. The first two steps of our construction mirror known biological networks, for example, in the fruit fly olfactory system [Caron et al., 2013; Lin et al., 2014; Dasgupta et al., 2017]. Our analysis helps provide a theoretical understanding of these networks and lay a foundation for how random compression and input memorization may be implemented in biological neural networks.
Technically, a contribution in our network design is the implementation of a short-term memory. Our network can be given a desired memory time t_m as an input parameter and satisfies the following with high probability: any pattern presented several times within a time window of t_m rounds will be mapped to a single representative output neuron. However, a pattern not presented for c⋅t_m rounds for some constant c>1 will be "forgotten", and its representative output neuron will be released, to accommodate newly introduced patterns.

Yael Hitron, Nancy Lynch, Cameron Musco, and Merav Parter. Random Sketching, Clustering, and Short-Term Memory in Spiking Neural Networks. In 11th Innovations in Theoretical Computer Science Conference (ITCS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 151, pp. 23:1-23:31, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{hitron_et_al:LIPIcs.ITCS.2020.23, author = {Hitron, Yael and Lynch, Nancy and Musco, Cameron and Parter, Merav}, title = {{Random Sketching, Clustering, and Short-Term Memory in Spiking Neural Networks}}, booktitle = {11th Innovations in Theoretical Computer Science Conference (ITCS 2020)}, pages = {23:1--23:31}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-134-4}, ISSN = {1868-8969}, year = {2020}, volume = {151}, editor = {Vidick, Thomas}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2020.23}, URN = {urn:nbn:de:0030-drops-117087}, doi = {10.4230/LIPIcs.ITCS.2020.23}, annote = {Keywords: biological distributed computing, spiking neural networks, compressed sensing, clustering, random projection, dimensionality reduction, winner-take-all} }

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**Published in:** LIPIcs, Volume 107, 45th International Colloquium on Automata, Languages, and Programming (ICALP 2018)

We give a simple distributed algorithm for computing adjacency matrix eigenvectors for the communication graph in an asynchronous gossip model. We show how to use this algorithm to give state-of-the-art asynchronous community detection algorithms when the communication graph is drawn from the well-studied stochastic block model. Our methods also apply to a natural alternative model of randomized communication, where nodes within a community communicate more frequently than nodes in different communities.
Our analysis simplifies and generalizes prior work by forging a connection between asynchronous eigenvector computation and Oja's algorithm for streaming principal component analysis. We hope that our work serves as a starting point for building further connections between the analysis of stochastic iterative methods, like Oja's algorithm, and work on asynchronous and gossip-type algorithms for distributed computation.

Frederik Mallmann-Trenn, Cameron Musco, and Christopher Musco. Eigenvector Computation and Community Detection in Asynchronous Gossip Models. In 45th International Colloquium on Automata, Languages, and Programming (ICALP 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 107, pp. 159:1-159:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{mallmanntrenn_et_al:LIPIcs.ICALP.2018.159, author = {Mallmann-Trenn, Frederik and Musco, Cameron and Musco, Christopher}, title = {{Eigenvector Computation and Community Detection in Asynchronous Gossip Models}}, booktitle = {45th International Colloquium on Automata, Languages, and Programming (ICALP 2018)}, pages = {159:1--159:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-076-7}, ISSN = {1868-8969}, year = {2018}, volume = {107}, editor = {Chatzigiannakis, Ioannis and Kaklamanis, Christos and Marx, D\'{a}niel and Sannella, Donald}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2018.159}, URN = {urn:nbn:de:0030-drops-91639}, doi = {10.4230/LIPIcs.ICALP.2018.159}, annote = {Keywords: block model, community detection, distributed clustering, eigenvector computation, gossip algorithms, population protocols} }

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

Understanding the singular value spectrum of an n x n matrix A is a fundamental task in countless numerical computation and data analysis applications. In matrix multiplication time, it is possible to perform a full SVD of A and directly compute the singular values \sigma_1,...,\sigma_n. However, little is known about algorithms that break this runtime barrier.
Using tools from stochastic trace estimation, polynomial approximation, and fast linear system solvers, we show how to efficiently isolate different ranges of A's spectrum and approximate the number of singular values in these ranges. We thus effectively compute an approximate histogram of the spectrum, which can stand in for the true singular values in many applications.
We use our histogram primitive to give the first algorithms for approximating a wide class of symmetric matrix norms and spectral sums faster than the best known runtime for matrix multiplication. For example, we show how to obtain a (1 + \epsilon) approximation to the Schatten 1-norm (i.e. the nuclear or trace norm) in just ~ O((nnz(A)n^{1/3} + n^2)\epsilon^{-3}) time for A with uniform row sparsity or \tilde O(n^{2.18} \epsilon^{-3}) time for dense matrices. The runtime scales smoothly for general Schatten-p norms, notably becoming \tilde O (p nnz(A) \epsilon^{-3}) for any real p >= 2.
At the same time, we show that the complexity of spectrum approximation is inherently tied to fast matrix multiplication in the small \epsilon regime. We use fine-grained complexity to give conditional lower bounds for spectrum approximation, showing that achieving milder \epsilon dependencies in our algorithms would imply triangle detection algorithms for general graphs running in faster than state of the art matrix multiplication time. This further implies, through a reduction of (Williams & William, 2010), that highly accurate spectrum approximation algorithms running in subcubic time can be used to give subcubic time matrix multiplication. As an application of our bounds, we show that precisely computing all effective resistances in a graph in less than matrix multiplication time is likely difficult, barring a major algorithmic breakthrough.

Cameron Musco, Praneeth Netrapalli, Aaron Sidford, Shashanka Ubaru, and David P. Woodruff. Spectrum Approximation Beyond Fast Matrix Multiplication: Algorithms and Hardness. In 9th Innovations in Theoretical Computer Science Conference (ITCS 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 94, pp. 8:1-8:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{musco_et_al:LIPIcs.ITCS.2018.8, author = {Musco, Cameron and Netrapalli, Praneeth and Sidford, Aaron and Ubaru, Shashanka and Woodruff, David P.}, title = {{Spectrum Approximation Beyond Fast Matrix Multiplication: Algorithms and Hardness}}, booktitle = {9th Innovations in Theoretical Computer Science Conference (ITCS 2018)}, pages = {8:1--8:21}, 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.2018.8}, URN = {urn:nbn:de:0030-drops-83397}, doi = {10.4230/LIPIcs.ITCS.2018.8}, annote = {Keywords: spectrum approximation, matrix norm computation, fine-grained complexity, linear algebra} }

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**Published in:** LIPIcs, Volume 67, 8th Innovations in Theoretical Computer Science Conference (ITCS 2017)

We initiate a line of investigation into biological neural networks from an algorithmic perspective. We develop a simplified but biologically plausible model for distributed computation in stochastic spiking neural networks and study tradeoffs between computation time and network complexity in this model. Our aim is to abstract real neural networks in a way that, while not capturing all interesting features, preserves high-level behavior and allows us to make biologically relevant conclusions.
In this paper, we focus on the important 'winner-take-all' (WTA) problem, which is analogous to a neural leader election unit: a network consisting of $n$ input neurons and n corresponding output neurons must converge to a state in which a single output corresponding to a firing input (the 'winner') fires, while all other outputs remain silent. Neural circuits for WTA rely on inhibitory neurons, which suppress the activity of competing outputs and drive the network towards a converged state with a single firing winner. We attempt to understand how the number of inhibitors used affects network convergence time.
We show that it is possible to significantly outperform naive WTA constructions through a more refined use of inhibition, solving the problem in O(\theta) rounds in expectation with just O(\log^{1/\theta} n) inhibitors for any \theta. An alternative construction gives convergence in O(\log^{1/\theta} n) rounds with O(\theta) inhibitors. We complement these upper bounds with our main technical contribution, a nearly matching lower bound for networks using \ge \log \log n inhibitors. Our lower bound uses familiar indistinguishability and locality arguments from distributed computing theory applied to the neural setting. It lets us derive a number of interesting conclusions about the structure of any network solving WTA with good probability, and the use of randomness and inhibition within such a network.

Nancy Lynch, Cameron Musco, and Merav Parter. Computational Tradeoffs in Biological Neural Networks: Self-Stabilizing Winner-Take-All Networks. In 8th Innovations in Theoretical Computer Science Conference (ITCS 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 67, pp. 15:1-15:44, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{lynch_et_al:LIPIcs.ITCS.2017.15, author = {Lynch, Nancy and Musco, Cameron and Parter, Merav}, title = {{Computational Tradeoffs in Biological Neural Networks: Self-Stabilizing Winner-Take-All Networks}}, booktitle = {8th Innovations in Theoretical Computer Science Conference (ITCS 2017)}, pages = {15:1--15:44}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-029-3}, ISSN = {1868-8969}, year = {2017}, volume = {67}, editor = {Papadimitriou, Christos H.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2017.15}, URN = {urn:nbn:de:0030-drops-81952}, doi = {10.4230/LIPIcs.ITCS.2017.15}, annote = {Keywords: biological distributed algorithms, neural networks, distributed lower bounds, winner-take-all networks} }

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**Published in:** LIPIcs, Volume 91, 31st International Symposium on Distributed Computing (DISC 2017)

We study distributed algorithms implemented in a simplified biologically inspired model for stochastic spiking neural networks. We focus on tradeoffs between computation time and network complexity, along with the role of noise and randomness in efficient neural computation.
It is widely accepted that neural spike responses, and neural computation in general, is inherently stochastic. In recent work, we explored how this stochasticity could be leveraged to solve the 'winner-take-all' leader election task. Here, we focus on using randomness in neural algorithms for similarity testing and compression. In the most basic setting, given two n-length patterns of firing neurons, we wish to distinguish if the patterns are equal or epsilon-far from equal.
Randomization allows us to solve this task with a very compact network, using O((sqrt(n) log n)/epsilon) auxiliary neurons, which is sublinear in the input size. At the heart of our solution is the design of a t-round neural random access memory, or indexing network, which we call a neuro-RAM. This module can be implemented with O(n/t) auxiliary neurons and is useful in many applications beyond similarity testing - e.g., we discuss its application to compression via random projection.
Using a VC dimension-based argument, we show that the tradeoff between runtime and network size in our neuro-RAM is nearly optimal. To the best of our knowledge, we are the first to apply these techniques to stochastic spiking networks. Our result has several implications - since our neuro-RAM can be implemented with deterministic threshold gates, it demonstrates that, in contrast to similarity testing, randomness does not provide significant computational advantages for this problem. It also establishes a separation between our networks, which spike with a sigmoidal probability function, and well-studied deterministic sigmoidal networks, whose gates output real number values, and which can implement a neuro-RAM much more efficiently.

Nancy Lynch, Cameron Musco, and Merav Parter. Neuro-RAM Unit with Applications to Similarity Testing and Compression in Spiking Neural Networks. In 31st International Symposium on Distributed Computing (DISC 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 91, pp. 33:1-33:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{lynch_et_al:LIPIcs.DISC.2017.33, author = {Lynch, Nancy and Musco, Cameron and Parter, Merav}, title = {{Neuro-RAM Unit with Applications to Similarity Testing and Compression in Spiking Neural Networks}}, booktitle = {31st International Symposium on Distributed Computing (DISC 2017)}, pages = {33:1--33:16}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-053-8}, ISSN = {1868-8969}, year = {2017}, volume = {91}, editor = {Richa, Andr\'{e}a}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.DISC.2017.33}, URN = {urn:nbn:de:0030-drops-79863}, doi = {10.4230/LIPIcs.DISC.2017.33}, annote = {Keywords: spiking neural networks, biological distributed algorithms, circuit design} }

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**Published in:** LIPIcs, Volume 60, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2016)

Finding a small spectral approximation for a tall n x d matrix A is a fundamental numerical primitive. For a number of reasons, one often seeks an approximation whose rows are sampled from those of A. Row sampling improves interpretability, saves space when A is sparse, and preserves row structure, which is especially important, for example, when A represents a graph.
However, correctly sampling rows from A can be costly when the matrix is large and cannot be stored and processed in memory. Hence, a number of recent publications focus on row sampling in the streaming setting, using little more space than what is required to store the outputted approximation [Kelner Levin 2013] [Kapralov et al. 2014].
Inspired by a growing body of work on online algorithms for machine learning and data analysis, we extend this work to a more restrictive online setting: we read rows of A one by one and immediately decide whether each row should be kept in the spectral approximation or discarded, without ever retracting these decisions. We present an extremely simple algorithm that approximates A up to multiplicative error epsilon and additive error delta using O(d log d log (epsilon ||A||_2^2/delta) / epsilon^2) online samples, with memory overhead proportional to the cost of storing the spectral approximation. We also present an algorithm that uses O(d^2) memory but only requires O(d log (epsilon ||A||_2^2/delta) / epsilon^2) samples, which we show is optimal.
Our methods are clean and intuitive, allow for lower memory usage than prior work, and expose new theoretical properties of leverage score based matrix approximation.

Michael B. Cohen, Cameron Musco, and Jakub Pachocki. Online Row Sampling. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 60, pp. 7:1-7:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{cohen_et_al:LIPIcs.APPROX-RANDOM.2016.7, author = {Cohen, Michael B. and Musco, Cameron and Pachocki, Jakub}, title = {{Online Row Sampling}}, booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2016)}, pages = {7:1--7:18}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-018-7}, ISSN = {1868-8969}, year = {2016}, volume = {60}, editor = {Jansen, Klaus and Mathieu, Claire and Rolim, Jos\'{e} D. P. and Umans, Chris}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX-RANDOM.2016.7}, URN = {urn:nbn:de:0030-drops-66304}, doi = {10.4230/LIPIcs.APPROX-RANDOM.2016.7}, annote = {Keywords: spectral sparsification, leverage score sampling, online sparsification} }