39,172 research outputs found

    Polynomials that Sign Represent Parity and Descartes' Rule of Signs

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    A real polynomial P(X1,...,Xn)P(X_1,..., X_n) sign represents f:An→{0,1}f: A^n \to \{0,1\} if for every (a1,...,an)∈An(a_1, ..., a_n) \in A^n, the sign of P(a1,...,an)P(a_1,...,a_n) equals (−1)f(a1,...,an)(-1)^{f(a_1,...,a_n)}. Such sign representations are well-studied in computer science and have applications to computational complexity and computational learning theory. In this work, we present a systematic study of tradeoffs between degree and sparsity of sign representations through the lens of the parity function. We attempt to prove bounds that hold for any choice of set AA. We show that sign representing parity over {0,...,m−1}n\{0,...,m-1\}^n with the degree in each variable at most m−1m-1 requires sparsity at least mnm^n. We show that a tradeoff exists between sparsity and degree, by exhibiting a sign representation that has higher degree but lower sparsity. We show a lower bound of n(m−2)+1n(m -2) + 1 on the sparsity of polynomials of any degree representing parity over {0,...,m−1}n\{0,..., m-1\}^n. We prove exact bounds on the sparsity of such polynomials for any two element subset AA. The main tool used is Descartes' Rule of Signs, a classical result in algebra, relating the sparsity of a polynomial to its number of real roots. As an application, we use bounds on sparsity to derive circuit lower bounds for depth-two AND-OR-NOT circuits with a Threshold Gate at the top. We use this to give a simple proof that such circuits need size 1.5n1.5^n to compute parity, which improves the previous bound of 4/3n/2{4/3}^{n/2} due to Goldmann (1997). We show a tight lower bound of 2n2^n for the inner product function over {0,1}n×{0,1}n\{0,1\}^n \times \{0, 1\}^n.Comment: To appear in Computational Complexit

    Polynomials with and without determinantal representations

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    The problem of writing real zero polynomials as determinants of linear matrix polynomials has recently attracted a lot of attention. Helton and Vinnikov have proved that any real zero polynomial in two variables has a determinantal representation. Br\"and\'en has shown that the result does not extend to arbitrary numbers of variables, disproving the generalized Lax conjecture. We prove that in fact almost no real zero polynomial admits a determinantal representation; there are dimensional differences between the two sets. So the generalized Lax conjecture fails badly. The result follows from a general upper bound on the size of linear matrix polynomials. We then provide a large class of surprisingly simple explicit real zero polynomials that do not have a determinantal representation, improving upon Br\"and\'en's mostly unconstructive result. We finally characterize polynomials of which some power has a determinantal representation, in terms of an algebra with involution having a finite dimensional representation. We use the characterization to prove that any quadratic real zero polynomial has a determinantal representation, after taking a high enough power. Taking powers is thereby really necessary in general. The representations emerge explicitly, and we characterize them up to unitary equivalence

    Discovery of statistical equivalence classes using computer algebra

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    Discrete statistical models supported on labelled event trees can be specified using so-called interpolating polynomials which are generalizations of generating functions. These admit a nested representation. A new algorithm exploits the primary decomposition of monomial ideals associated with an interpolating polynomial to quickly compute all nested representations of that polynomial. It hereby determines an important subclass of all trees representing the same statistical model. To illustrate this method we analyze the full polynomial equivalence class of a staged tree representing the best fitting model inferred from a real-world dataset.Comment: 26 pages, 9 figure

    Limits on Representing Boolean Functions by Linear Combinations of Simple Functions: Thresholds, ReLUs, and Low-Degree Polynomials

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    We consider the problem of representing Boolean functions exactly by "sparse" linear combinations (over R\mathbb{R}) of functions from some "simple" class C{\cal C}. In particular, given C{\cal C} we are interested in finding low-complexity functions lacking sparse representations. When C{\cal C} is the set of PARITY functions or the set of conjunctions, this sort of problem has a well-understood answer, the problem becomes interesting when C{\cal C} is "overcomplete" and the set of functions is not linearly independent. We focus on the cases where C{\cal C} is the set of linear threshold functions, the set of rectified linear units (ReLUs), and the set of low-degree polynomials over a finite field, all of which are well-studied in different contexts. We provide generic tools for proving lower bounds on representations of this kind. Applying these, we give several new lower bounds for "semi-explicit" Boolean functions. For example, we show there are functions in nondeterministic quasi-polynomial time that require super-polynomial size: ∙\bullet Depth-two neural networks with sign activation function, a special case of depth-two threshold circuit lower bounds. ∙\bullet Depth-two neural networks with ReLU activation function. ∙\bullet R\mathbb{R}-linear combinations of O(1)O(1)-degree Fp\mathbb{F}_p-polynomials, for every prime pp (related to problems regarding Higher-Order "Uncertainty Principles"). We also obtain a function in ENPE^{NP} requiring 2Ω(n)2^{\Omega(n)} linear combinations. ∙\bullet R\mathbb{R}-linear combinations of ACC∘THRACC \circ THR circuits of polynomial size (further generalizing the recent lower bounds of Murray and the author). (The above is a shortened abstract. For the full abstract, see the paper.

    Rational Convolution Roots of Isobaric Polynomials

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    In this paper, we exhibit two matrix representations of the rational roots of generalized Fibonacci polynomials (GFPs) under convolution product, in terms of determinants and permanents, respectively. The underlying root formulas for GFPs and for weighted isobaric polynomials (WIPs), which appeared in an earlier paper by MacHenry and Tudose, make use of two types of operators. These operators are derived from the generating functions for Stirling numbers of the first kind and second kind. Hence we call them Stirling operators. To construct matrix representations of the roots of GFPs, we use the Stirling operators of the first kind. We give explicit examples to show how the Stirling operators of the second kind appear in the low degree cases for the WIP-roots. As a consequence of the matrix construction, we have matrix representations of multiplicative arithmetic functions under the Dirichlet product into its divisible closure.Comment: 13 page

    Certified Roundoff Error Bounds using Bernstein Expansions and Sparse Krivine-Stengle Representations

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    Floating point error is an inevitable drawback of embedded systems implementation. Computing rigorous upper bounds of roundoff errors is absolutely necessary to the validation of critical software. This problem is even more challenging when addressing non-linear programs. In this paper, we propose and compare two new methods based on Bernstein expansions and sparse Krivine-Stengle representations, adapted from the field of the global optimization to compute upper bounds of roundoff errors for programs implementing polynomial functions. We release two related software package FPBern and FPKiSten, and compare them with state of the art tools. We show that these two methods achieve competitive performance, while computing accurate upper bounds by comparison with other tools.Comment: 20 pages, 2 table
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