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**Published in:** LIPIcs, Volume 215, 13th Innovations in Theoretical Computer Science Conference (ITCS 2022)

One-bit compressed sensing (1bCS) is an extreme-quantized signal acquisition method that has been intermittently studied in the past decade. In 1bCS, linear samples of a high dimensional signal are quantized to only one bit per sample (sign of the measurement). The extreme quantization makes it an interesting case study of the more general single-index or generalized linear models. At the same time it can also be thought of as a "design" version of learning a binary linear classifier or halfspace-learning.
Assuming the original signal vector to be sparse, existing results in 1bCS either aim to find the support of the vector, or approximate the signal within an ε-ball. The focus of this paper is support recovery, which often also computationally facilitate approximate signal recovery. A universal measurement matrix for 1bCS refers to one set of measurements that work for all sparse signals. With universality, it is known that Θ̃(k²) 1bCS measurements are necessary and sufficient for support recovery (where k denotes the sparsity). In this work, we show that it is possible to universally recover the support with a small number of false positives with Õ(k^{3/2}) measurements. If the dynamic range of the signal vector is known, then with a different technique, this result can be improved to only Õ(k) measurements. Other results on universal but approximate support recovery are also provided in this paper. All of our main recovery algorithms are simple and polynomial-time.

Arya Mazumdar and Soumyabrata Pal. Support Recovery in Universal One-Bit Compressed Sensing. In 13th Innovations in Theoretical Computer Science Conference (ITCS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 215, pp. 106:1-106:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{mazumdar_et_al:LIPIcs.ITCS.2022.106, author = {Mazumdar, Arya and Pal, Soumyabrata}, title = {{Support Recovery in Universal One-Bit Compressed Sensing}}, booktitle = {13th Innovations in Theoretical Computer Science Conference (ITCS 2022)}, pages = {106:1--106:20}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-217-4}, ISSN = {1868-8969}, year = {2022}, volume = {215}, editor = {Braverman, Mark}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2022.106}, URN = {urn:nbn:de:0030-drops-157028}, doi = {10.4230/LIPIcs.ITCS.2022.106}, annote = {Keywords: Superset Recovery, Approximate Support Recovery, List union-free family, Descartes’ rule of signs} }

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RANDOM

**Published in:** LIPIcs, Volume 145, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2019)

Random geometric graph (Gilbert, 1961) is a basic model of random graphs for spatial networks proposed shortly after the introduction of the Erdős-Rényi random graphs. The geometric block model (GBM) is a probabilistic model for community detection defined over random geometric graphs (RGG) similar in spirit to the popular stochastic block model which is defined over Erdős-Rényi random graphs. The GBM naturally inherits many desirable properties of RGGs such as transitivity ("friends having common friends') and has been shown to model many real-world networks better than the stochastic block model. Analyzing the properties of a GBM requires new tools and perspectives to handle correlation in edge formation. In this paper, we study the necessary and sufficient conditions for community recovery over GBM in the connectivity regime. We provide efficient algorithms that recover the communities exactly with high probability and match the lower bound within a small constant factor. This requires us to prove new connectivity results for vertex-random graphs or random annulus graphs which are natural generalizations of random geometric graphs.
A vertex-random graph is a model of random graphs where the randomness lies in the vertices as opposed to an Erdős-Rényi random graph where the randomness lies in the edges. A vertex-random graph G(n, [r_1, r_2]), 0 <=r_1 <r_2 <=1 with n vertices is defined by assigning a real number in [0,1] randomly and uniformly to each vertices and adding an edge between two vertices if the "distance" between the corresponding two random numbers is between r_1 and r_2. For the special case of r_1=0, this corresponds to random geometric graph in one dimension. We can extend this model naturally to higher dimensions; these higher dimensional counterparts are referred to as random annulus graphs. Random annulus graphs appear naturally whenever the well-known Goldilocks principle ("not too close, not too far') holds in a network. In this paper, we study the connectivity properties of such graphs, providing both necessary and sufficient conditions. We show a surprising long edge phenomena for vertex-random graphs: the minimum gap for connectivity between r_1 and r_2 is significantly less when r_1 >0 vs when r_1=0 (RGG). We then extend the connectivity results to high dimensions. These results play a crucial role in analyzing the GBM.

Sainyam Galhotra, Arya Mazumdar, Soumyabrata Pal, and Barna Saha. Connectivity of Random Annulus Graphs and the Geometric Block Model. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 145, pp. 53:1-53:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{galhotra_et_al:LIPIcs.APPROX-RANDOM.2019.53, author = {Galhotra, Sainyam and Mazumdar, Arya and Pal, Soumyabrata and Saha, Barna}, title = {{Connectivity of Random Annulus Graphs and the Geometric Block Model}}, booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2019)}, pages = {53:1--53:23}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-125-2}, ISSN = {1868-8969}, year = {2019}, volume = {145}, editor = {Achlioptas, Dimitris and V\'{e}gh, L\'{a}szl\'{o} A.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX-RANDOM.2019.53}, URN = {urn:nbn:de:0030-drops-112682}, doi = {10.4230/LIPIcs.APPROX-RANDOM.2019.53}, annote = {Keywords: random graphs, geometric graphs, community detection, block model} }

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**Published in:** LIPIcs, Volume 144, 27th Annual European Symposium on Algorithms (ESA 2019)

In the beautifully simple-to-state problem of trace reconstruction, the goal is to reconstruct an unknown binary string x given random "traces" of x where each trace is generated by deleting each coordinate of x independently with probability p<1. The problem is well studied both when the unknown string is arbitrary and when it is chosen uniformly at random. For both settings, there is still an exponential gap between upper and lower sample complexity bounds and our understanding of the problem is still surprisingly limited. In this paper, we consider natural parameterizations and generalizations of this problem in an effort to attain a deeper and more comprehensive understanding. Perhaps our most surprising results are:
1) We prove that exp(O(n^(1/4) sqrt{log n})) traces suffice for reconstructing arbitrary matrices. In the matrix version of the problem, each row and column of an unknown sqrt{n} x sqrt{n} matrix is deleted independently with probability p. Our results contrasts with the best known results for sequence reconstruction where the best known upper bound is exp(O(n^(1/3))).
2) An optimal result for random matrix reconstruction: we show that Theta(log n) traces are necessary and sufficient. This is in contrast to the problem for random sequences where there is a super-logarithmic lower bound and the best known upper bound is exp({O}(log^(1/3) n)).
3) We show that exp(O(k^(1/3) log^(2/3) n)) traces suffice to reconstruct k-sparse strings, providing an improvement over the best known sequence reconstruction results when k = o(n/log^2 n).
4) We show that poly(n) traces suffice if x is k-sparse and we additionally have a "separation" promise, specifically that the indices of 1’s in x all differ by Omega(k log n).

Akshay Krishnamurthy, Arya Mazumdar, Andrew McGregor, and Soumyabrata Pal. Trace Reconstruction: Generalized and Parameterized. In 27th Annual European Symposium on Algorithms (ESA 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 144, pp. 68:1-68:25, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{krishnamurthy_et_al:LIPIcs.ESA.2019.68, author = {Krishnamurthy, Akshay and Mazumdar, Arya and McGregor, Andrew and Pal, Soumyabrata}, title = {{Trace Reconstruction: Generalized and Parameterized}}, booktitle = {27th Annual European Symposium on Algorithms (ESA 2019)}, pages = {68:1--68:25}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-124-5}, ISSN = {1868-8969}, year = {2019}, volume = {144}, editor = {Bender, Michael A. and Svensson, Ola 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.2019.68}, URN = {urn:nbn:de:0030-drops-111891}, doi = {10.4230/LIPIcs.ESA.2019.68}, annote = {Keywords: deletion channel, trace reconstruction, matrix reconstruction} }

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