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1. Ellipsoid Fitting Up to A Constant

   Junting Hsieh, Pravesh K (July 2023, International Colloquium on Automata, Languages and Programming, ICALP)

   In [Saunderson, 2011; Saunderson et al., 2013], Saunderson, Parrilo, and Willsky asked the following elegant geometric question: what is the largest m = m(d) such that there is an ellipsoid in ℝ^d that passes through v_1, v_2, …, v_m with high probability when the v_is are chosen independently from the standard Gaussian distribution N(0,I_d)? The existence of such an ellipsoid is equivalent to the existence of a positive semidefinite matrix X such that v_i^⊤ X v_i = 1 for every 1 ⩽ i ⩽ m - a natural example of a random semidefinite program. SPW conjectured that m = (1-o(1)) d²/4 with high probability. Very recently, Potechin, Turner, Venkat and Wein [Potechin et al., 2022] and Kane and Diakonikolas [Kane and Diakonikolas, 2022] proved that m ≳ d²/log^O(1) d via a certain natural, explicit construction. In this work, we give a substantially tighter analysis of their construction to prove that m ≳ d²/C for an absolute constant C > 0. This resolves one direction of the SPW conjecture up to a constant. Our analysis proceeds via the method of Graphical Matrix Decomposition that has recently been used to analyze correlated random matrices arising in various areas [Barak et al., 2019; Bafna et al., 2022]. Our key new technical tool is a refined method to prove singular value upper bounds on certain correlated random matrices that are tight up to absolute dimension-independent constants. In contrast, all previous methods that analyze such matrices lose logarithmic factors in the dimension. 

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2. Near-optimal fitting of ellipsoids to random points

   Potechin, Aaron; Turner, Paxton M.; Venkat, Prayaag; Wein, Alexander S. (July 2023, Proceedings of Machine Learning Research)

   Given independent standard Gaussian points v1, . . . , vn in dimension d, for what values of (n, d) does there exist with high probability an origin-symmetric ellipsoid that simultaneously passes through all of the points? This basic problem of fitting an ellipsoid to random points has connections to low-rank matrix decompositions, independent component analysis, and principal component analysis. Based on strong numerical evidence, Saunderson, Parrilo, and Willsky (Saunderson, 2011; Saunderson et al., 2013) conjectured that the ellipsoid fitting problem transitions from feasible to infeasible as the number of points n increases, with a sharp threshold at n ∼ d^2/4. We resolve this conjecture up to logarithmic factors by constructing a fitting ellipsoid for some n = d^2/polylog(d). Our proof demonstrates feasibility of the least squares construction of (Saunderson, 2011; Saunderson et al., 2013) using a convenient decomposition of a certain non-standard random matrix and a careful analysis of its Neumann expansion via the theory of graph matrices. 

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3. A nearly-optimal bound for fast regression with ℓ∞ guarantee

   Song, Zhao; Ye, Mingquan; Yin, Junze; Zhang, Lichen (July 2023, Proceedings of Machine Learning Research)

   Given a matrix A ∈ ℝn\texttimes{}d and a vector b ∈ ℝn, we consider the regression problem with ℓ∞ guarantees: finding a vector x′ ∈ ℝd such that $$||x'-x^* ||_infty leq frac{epsilon}{sqrt{d}}cdot ||Ax^*-b||_2cdot ||A^dagger||$$, where x* = arg minx∈Rd ||Ax – b||2. One popular approach for solving such ℓ2 regression problem is via sketching: picking a structured random matrix S ∈ ℝm\texttimes{}n with m < n and S A can be quickly computed, solve the "sketched" regression problem arg minx∈ℝd ||S Ax – Sb||2. In this paper, we show that in order to obtain such ℓ∞ guarantee for ℓ2 regression, one has to use sketching matrices that are dense. To the best of our knowledge, this is the first user case in which dense sketching matrices are necessary. On the algorithmic side, we prove that there exists a distribution of dense sketching matrices with m = ε-2d log3(n/δ) such that solving the sketched regression problem gives the ℓ∞ guarantee, with probability at least 1 – δ. Moreover, the matrix S A can be computed in time O(nd log n). Our row count is nearly-optimal up to logarithmic factors, and significantly improves the result in (Price et al., 2017), in which a superlinear in d rows, m = Ω(ε-2d1+γ) for γ ∈ (0, 1) is required. Moreover, we develop a novel analytical framework for ℓ∞ guarantee regression that utilizes the Oblivious Coordinate-wise Embedding (OCE) property introduced in (Song \& Yu, 2021). Our analysis is much simpler and more general than that of (Price et al., 2017). Leveraging this framework, we extend the ℓ∞ guarantee regression result to dense sketching matrices for computing the fast tensor product of vectors. 

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4. Leibniz International Proceedings in Informatics (LIPIcs):50th International Colloquium on Automata, Languages, and Programming (ICALP 2023)

   https://doi.org/10.4230/LIPIcs.ICALP.2023.41  

   Cheng, Kuan; Jin, Zhengzhong; Li, Xin; Wei, Zhide; Zheng, Yu (January 2023, Schloss Dagstuhl – Leibniz-Zentrum für Informatik)

   Etessami, Kousha; Feige, Uriel; Puppis, Gabriele (Ed.)

   This work continues the study of linear error correcting codes against adversarial insertion deletion errors (insdel errors). Previously, the work of Cheng, Guruswami, Haeupler, and Li [Kuan Cheng et al., 2021] showed the existence of asymptotically good linear insdel codes that can correct arbitrarily close to 1 fraction of errors over some constant size alphabet, or achieve rate arbitrarily close to 1/2 even over the binary alphabet. As shown in [Kuan Cheng et al., 2021], these bounds are also the best possible. However, known explicit constructions in [Kuan Cheng et al., 2021], and subsequent improved constructions by Con, Shpilka, and Tamo [Con et al., 2022] all fall short of meeting these bounds. Over any constant size alphabet, they can only achieve rate < 1/8 or correct < 1/4 fraction of errors; over the binary alphabet, they can only achieve rate < 1/1216 or correct < 1/54 fraction of errors. Apparently, previous techniques face inherent barriers to achieve rate better than 1/4 or correct more than 1/2 fraction of errors. In this work we give new constructions of such codes that meet these bounds, namely, asymptotically good linear insdel codes that can correct arbitrarily close to 1 fraction of errors over some constant size alphabet, and binary asymptotically good linear insdel codes that can achieve rate arbitrarily close to 1/2. All our constructions are efficiently encodable and decodable. Our constructions are based on a novel approach of code concatenation, which embeds the index information implicitly into codewords. This significantly differs from previous techniques and may be of independent interest. Finally, we also prove the existence of linear concatenated insdel codes with parameters that match random linear codes, and propose a conjecture about linear insdel codes. 

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5. Almost Chor-Goldreich Sources and Adversarial Random Walks

   https://doi.org/10.1145/3564246.3585134  

   Doron, Dean; Moshkovitz, Dana; Oh, Justin; Zuckerman, David (June 2023, Proceedings of the annual ACM Symposium on Theory of Computing)

   A Chor–Goldreich (CG) source is a sequence of random variables X = X1 ∘ … ∘ Xt, where each Xi ∼ {0,1}d and Xi has δ d min-entropy conditioned on any fixing of X1 ∘ … ∘ Xi−1. The parameter 0<δ≤ 1 is the entropy rate of the source. We typically think of d as constant and t as growing. We extend this notion in several ways, defining almost CG sources. Most notably, we allow each Xi to only have conditional Shannon entropy δ d. We achieve pseudorandomness results for almost CG sources which were not known to hold even for standard CG sources, and even for the weaker model of Santha–Vazirani sources: We construct a deterministic condenser that on input X, outputs a distribution which is close to having constant entropy gap, namely a distribution Z ∼ {0,1}m for m ≈ δ dt with min-entropy m−O(1). Therefore, we can simulate any randomized algorithm with small failure probability using almost CG sources with no multiplicative slowdown. This result extends to randomized protocols as well, and any setting in which we cannot simply cycle over all seeds, and a “one-shot” simulation is needed. Moreover, our construction works in an online manner, since it is based on random walks on expanders. Our main technical contribution is a novel analysis of random walks, which should be of independent interest. We analyze walks with adversarially correlated steps, each step being entropy-deficient, on good enough lossless expanders. We prove that such walks (or certain interleaved walks on two expanders), starting from a fixed vertex and walking according to X1∘ … ∘ Xt, accumulate most of the entropy in X. 

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