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1. Dimension Reduction for Polynomials over Gaussian Space and Applications

   Ghazi, B. ; Kamath, P. ; Raghavendra, P. ( June 2018 , Computational Complexity Conference (CCC))

   We introduce a new technique for reducing the dimension of the ambient space of low-degree polynomials in the Gaussian space while preserving their relative correlation structure. As an application, we obtain an explicit upper bound on the dimension of an epsilon-optimal noise-stable Gaussian partition. In fact, we address the more general problem of upper bounding the number of samples needed to epsilon-approximate any joint distribution that can be non-interactively simulated from a correlated Gaussian source. Our results significantly improve (from Ackermann-like to "merely" exponential) the upper bounds recently proved on the above problems by De, Mossel & Neeman [CCC 2017, SODA 2018 resp.] and imply decidability of the larger alphabet case of the gap non-interactive simulation problem posed by Ghazi, Kamath & Sudan [FOCS 2016]. Our technique of dimension reduction for low-degree polynomials is simple and can be seen as a generalization of the Johnson-Lindenstrauss lemma and could be of independent interest.
2. Monotone circuit lower bounds from resolution

   Garg, Ankit ; Goos, Mika ; Kamath, Pritish ; Sokolov, Dmitry ( June 2018 , Proceedings of the annual ACM Symposium on Theory of Computing)

   We introduce a new technique for reducing the dimension of the ambient space of low-degree polynomials in the Gaussian space while preserving their relative correlation structure. As an application, we obtain an explicit upper bound on the dimension of an epsilon-optimal noise-stable Gaussian partition. In fact, we address the more general problem of upper bounding the number of samples needed to epsilon-approximate any joint distribution that can be non-interactively simulated from a correlated Gaussian source. Our results significantly improve (from Ackermann-like to "merely" exponential) the upper bounds recently proved on the above problems by De, Mossel & Neeman [CCC 2017, SODA 2018 resp.] and imply decidability of the larger alphabet case of the gap non-interactive simulation problem posed by Ghazi, Kamath & Sudan [FOCS 2016]. Our technique of dimension reduction for low-degree polynomials is simple and can be seen as a generalization of the Johnson-Lindenstrauss lemma and could be of independent interest.
3. Proof Complexity Lower Bounds from Algebraic Circuit Complexity

   https://doi.org/10.4086/toc.2021.v017a010  

   Forbes, Michael A. ; Shpilka, Amir ; Tzameret, Iddo ; Wigderson, Avi ( January 2021 , Theory of Computing)

   Raz, Ran (Ed.)

   We give upper and lower bounds on the power of subsystems of the Ideal Proof System (IPS), the algebraic proof system recently proposed by Grochow and Pitassi, where the circuits comprising the proof come from various restricted algebraic circuit classes. This mimics an established research direction in the boolean setting for subsystems of Extended Frege proofs whose lines are circuits from restricted boolean circuit classes. Essentially all of the subsystems considered in this paper can simulate the well-studied Nullstellensatz proof system, and prior to this work there were no known lower bounds when measuring proof size by the algebraic complexity of the polynomials (except with respect to degree, or to sparsity). Our main contributions are two general methods of converting certain algebraic lower bounds into proof complexity ones. Both require stronger arithmetic lower bounds than common, which should hold not for a specific polynomial but for a whole family defined by it. These may be likened to some of the methods by which Boolean circuit lower bounds are turned into related proof-complexity ones, especially the "feasible interpolation" technique. We establish algebraic lower bounds of these forms for several explicit polynomials, against a variety of classes, and infer the relevant proofmore »complexity bounds. These yield separations between IPS subsystems, which we complement by simulations to create a partial structure theory for IPS systems. Our first method is a functional lower bound, a notion of Grigoriev and Razborov, which is a function f' from n-bit strings to a field, such that any polynomial f agreeing with f' on the boolean cube requires large algebraic circuit complexity. We develop functional lower bounds for a variety of circuit classes (sparse polynomials, depth-3 powering formulas, read-once algebraic branching programs and multilinear formulas) where f'(x) equals 1/p(x) for a constant-degree polynomial p depending on the relevant circuit class. We believe these lower bounds are of independent interest in algebraic complexity, and show that they also imply lower bounds for the size of the corresponding IPS refutations for proving that the relevant polynomial p is non-zero over the boolean cube. In particular, we show super-polynomial lower bounds for refuting variants of the subset-sum axioms in these IPS subsystems. Our second method is to give lower bounds for multiples, that is, to give explicit polynomials whose all (non-zero) multiples require large algebraic circuit complexity. By extending known techniques, we give lower bounds for multiples for various restricted circuit classes such sparse polynomials, sums of powers of low-degree polynomials, and roABPs. These results are of independent interest, as we argue that lower bounds for multiples is the correct notion for instantiating the algebraic hardness versus randomness paradigm of Kabanets and Impagliazzo. Further, we show how such lower bounds for multiples extend to lower bounds for refutations in the corresponding IPS subsystem.« less
4. The Large-Error Approximate Degree of AC0

   https://doi.org/10.4230/LIPIcs.APPROX-RANDOM.2019.55  

   Bun, Mark ; Thaler, Justin ( September 2019 , Leibniz international proceedings in informatics)

   We prove two new results about the inability of low-degree polynomials to uniformly approximate constant-depth circuits, even to slightly-better-than-trivial error. First, we prove a tight Omega~(n^{1/2}) lower bound on the threshold degree of the SURJECTIVITY function on n variables. This matches the best known threshold degree bound for any AC^0 function, previously exhibited by a much more complicated circuit of larger depth (Sherstov, FOCS 2015). Our result also extends to a 2^{Omega~(n^{1/2})} lower bound on the sign-rank of an AC^0 function, improving on the previous best bound of 2^{Omega(n^{2/5})} (Bun and Thaler, ICALP 2016). Second, for any delta>0, we exhibit a function f : {-1,1}^n -> {-1,1} that is computed by a circuit of depth O(1/delta) and is hard to approximate by polynomials in the following sense: f cannot be uniformly approximated to error epsilon=1-2^{-Omega(n^{1-delta})}, even by polynomials of degree n^{1-delta}. Our recent prior work (Bun and Thaler, FOCS 2017) proved a similar lower bound, but which held only for error epsilon=1/3. Our result implies 2^{Omega(n^{1-delta})} lower bounds on the complexity of AC^0 under a variety of basic measures such as discrepancy, margin complexity, and threshold weight. This nearly matches the trivial upper bound of 2^{O(n)} that holds for everymore »function. The previous best lower bound on AC^0 for these measures was 2^{Omega(n^{1/2})} (Sherstov, FOCS 2015). Additional applications in learning theory, communication complexity, and cryptography are described.« less
5. Plücker coordinates of the best-fit Stiefel tropical linear space to a mixture of Gaussian distributions

   https://doi.org/10.1007/s41884-023-00098-w  

   Miura, Keiji ; Yoshida, Ruriko ( January 2023 , Information Geometry)

   In this research, we investigate a tropical principal component analysis (PCA) as a best-fit Stiefel tropical linear space to a given sample over the tropical projective torus for its dimensionality reduction and visualization. Especially, we characterize the best-fit Stiefel tropical linear space to a sample generated from a mixture of Gaussian distributions as the variances of the Gaussians go to zero. For a single Gaussian distribution, we show that the sum of residuals in terms of the tropical metric with the max-plus algebra over a given sample to a fitted Stiefel tropical linear space converges to zero by giving an upper bound for its convergence rate. Meanwhile, for a mixtures of Gaussian distribution, we show that the best-fit tropical linear space can be determined uniquely when we send variances to zero. We briefly consider the best-fit topical polynomial as an extension for the mixture of more than two Gaussians over the tropical projective space of dimension three. We show some geometric properties of these tropical linear spaces and polynomials.