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1. Efficient Algorithms for Least Square Piecewise Polynomial Regression

   Lokshtanov, Daniel ; Suri, Subhash ; Xue, Jie ( September 2021 , ESA21: Proceedings of European Symposium on Algorithms)

   null (Ed.)

   We present approximation and exact algorithms for piecewise regression of univariate and bivariate data using fixed-degree polynomials. Specifically, given a set S of n data points (x1, y1), . . . , (xn, yn) ∈ Rd × R where d ∈ {1, 2}, the goal is to segment xi’s into some (arbitrary) number of disjoint pieces P1, . . . , Pk, where each piece Pj is associated with a fixed-degree polynomial fj : Rd → R, to minimize the total loss function λk+􏰄ni=1(yi −f(xi))2, where λ ≥ 0 is a regularization term that penalizes model complexity (number of pieces) and f : 􏰇kj=1 Pj → R is the piecewise polynomial function defined as f|Pj = fj. The pieces P1,...,Pk are disjoint intervals of R in the case of univariate data and disjoint axis-aligned rectangles in the case of bivariate data. Our error approximation allows use of any fixed-degree polynomial, not just linear functions. Our main results are the following. For univariate data, we present a (1 + ε)-approximation algorithm with time complexity O(nε log1ε), assuming that data is presented in sorted order of xi’s. For bivariate data, we √ present three results: a sub-exponential exact algorithm with running time nO( n); a polynomial-time constant- approximation algorithm; and a quasi-polynomial time approximation scheme (QPTAS). The bivariate case is believed to be NP-hard in the folklore but we could not find a published record in the literature, so in this paper we also present a hardness proof for completeness. 

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2. Introducing isodynamic points for binary forms and their ratios

   https://doi.org/10.1007/s40627-022-00112-4  

   Hägg, Christian ; Shapiro, Boris ; Shapiro, Michael ( January 2023 , Complex Analysis and its Synergies)

   The isodynamic points of a plane triangle are known to be the only pair of its centers invariant under the action of the Möbius group$${\mathcal {M}}$$$M$on the set of triangles, Kimberling (Encyclopedia of Triangle Centers,http://faculty.evansville.edu/ck6/encyclopedia). Generalizing this classical result, we introduce below theisodynamicmap associating to a univariate polynomial of degree$$d\ge 3$$$d\ge 3$with at most double roots a polynomial of degree (at most)$$2d-4$$$2d-4$such that this map commutes with the action of the Möbius group$${\mathcal {M}}$$$M$on the zero loci of the initial polynomial and its image. The roots of the image polynomial will be called theisodynamic pointsof the preimage polynomial. Our construction naturally extends from univariate polynomials to binary forms and further to their ratios.

    

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3. Some bounds arising from a polynomial ideal associated to any $t$-design

   https://doi.org/10.13069/jacodesmath.729446  

   MARTIN, William J ; STINSON, Douglas R ( May 2020 , Journal of Algebra Combinatorics Discrete Structures and Applications)

   null (Ed.)

   We consider ordered pairs (X,B) where X is a finite set of size v and B is some collection of k-element subsets of X such that every t-element subset of X is contained in exactly λ "blocks'' b ∈B for some fixed λ. We represent each block b by a zero-one vector c_b of length v and explore the ideal I(B) of polynomials in v variables with complex coefficients which vanish on the set { c_b ∣ b ∈ B}. After setting up the basic theory, we investigate two parameters related to this ideal: γ1(B) is the smallest degree of a non-trivial polynomial in the ideal I(B) and γ2(B) is the smallest integer s such that I(B) is generated by a set of polynomials of degree at most s. We first prove the general bounds t/2 < γ1(B) ≤ γ2(B) ≤ k. Examining important families of examples, we find that, for symmetric 2-designs and Steiner systems, we have γ2(B) ≤ t. But we expect γ2(B) to be closer to k for less structured designs and we indicate this by constructing infinitely many triple systems satisfying γ2(B) = k. 

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4. 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 proof 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. 

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5. Low-Rank Univariate Sum of Squares Has No Spurious Local Minima

   https://doi.org/10.1137/22M1516208  

   Legat, Benoît ; Yuan, Chenyang ; Parrilo, Pablo ( September 2023 , SIAM Journal on Optimization)

   We study the problem of decomposing a polynomial p into a sum of r squares by minimizing a quadratically penalized objective fp(u)=‖‖∑ri=1u2i−p‖‖2. This objective is nonconvex and is equivalent to the rank-r Burer-Monteiro factorization of a semidefinite program (SDP) encoding the sum of squares decomposition. We show that for all univariate polynomials p, if r≥2 then fp(u) has no spurious second-order critical points, showing that all local optima are also global optima. This is in contrast to previous work showing that for general SDPs, in addition to genericity conditions, r has to be roughly the square root of the number of constraints (the degree of p) for there to be no spurious second-order critical points. Our proof uses tools from computational algebraic geometry and can be interpreted as constructing a certificate using the first- and second-order necessary conditions. We also show that by choosing a norm based on sampling equally-spaced points on the circle, the gradient ∇fp can be computed in nearly linear time using fast Fourier transforms. Experimentally we demonstrate that this method has very fast convergence using first-order optimization algorithms such as L-BFGS, with near-linear scaling to million-degree polynomials. 

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