39,172 research outputs found
Polynomials that Sign Represent Parity and Descartes' Rule of Signs
A real polynomial sign represents if
for every , the sign of equals
. 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
. We show that sign representing parity over with the
degree in each variable at most requires sparsity at least . 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 on the sparsity of polynomials of any degree representing
parity over . We prove exact bounds on the sparsity of such
polynomials for any two element subset . 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 to compute parity, which improves the previous bound
of due to Goldmann (1997). We show a tight lower bound of
for the inner product function over .Comment: To appear in Computational Complexit
Polynomials with and without determinantal representations
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
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
We consider the problem of representing Boolean functions exactly by "sparse"
linear combinations (over ) of functions from some "simple" class
. In particular, given we are interested in finding
low-complexity functions lacking sparse representations. When 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 is
"overcomplete" and the set of functions is not linearly independent. We focus
on the cases where 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:
Depth-two neural networks with sign activation function, a special
case of depth-two threshold circuit lower bounds.
Depth-two neural networks with ReLU activation function.
-linear combinations of -degree
-polynomials, for every prime (related to problems regarding
Higher-Order "Uncertainty Principles"). We also obtain a function in
requiring linear combinations.
-linear combinations of 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
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
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
- âŠ