38,471 research outputs found
On Higher-Order Fourier Analysis over Non-Prime Fields
The celebrated Weil bound for character sums says that for any low-degree polynomial P and any additive character chi, either chi(P) is a constant function or it is distributed close to uniform. The goal of higher-order Fourier analysis is to understand the connection between the algebraic and analytic properties of polynomials (and functions, generally) at a more detailed level. For instance, what is the tradeoff between the equidistribution of chi(P) and its "structure"?
Previously, most of the work in this area was over fields of prime order. We extend the tools of higher-order Fourier analysis to analyze functions over general finite fields. Let K be a field extension of a prime finite field F_p. Our technical results are:
1. If P: K^n -> K is a polynomial of degree |K|^{-s} for some s > 0 and non-trivial additive character chi, then P is a function of O_{d, s}(1) many non-classical polynomials of weight degree < d. The definition of non-classical polynomials over non-prime fields is one of the contributions of this work.
2. Suppose K and F are of bounded order, and let H be an affine subspace of K^n. Then, if P: K^n -> K is a polynomial of degree d that is sufficiently regular, then (P(x): x in H) is distributed almost as uniformly as possible subject to constraints imposed by the degree of P. Such a theorem was previously known for H an affine subspace over a prime field.
The tools of higher-order Fourier analysis have found use in different areas of computer science, including list decoding, algorithmic decomposition and testing. Using our new results, we revisit some of these areas.
(i) For any fixed finite field K, we show that the list decoding radius of the generalized Reed Muller code over K equals the minimum distance of the code.
(ii) For any fixed finite field K, we give a polynomial time algorithm to decide whether a given polynomial P: K^n -> K can be decomposed as a particular composition of lesser degree polynomials.
(iii) For any fixed finite field K, we prove that all locally characterized affine-invariant properties of functions f: K^n -> K are testable with one-sided error
Low-degree tests at large distances
We define tests of boolean functions which distinguish between linear (or
quadratic) polynomials, and functions which are very far, in an appropriate
sense, from these polynomials. The tests have optimal or nearly optimal
trade-offs between soundness and the number of queries.
In particular, we show that functions with small Gowers uniformity norms
behave ``randomly'' with respect to hypergraph linearity tests.
A central step in our analysis of quadraticity tests is the proof of an
inverse theorem for the third Gowers uniformity norm of boolean functions.
The last result has also a coding theory application. It is possible to
estimate efficiently the distance from the second-order Reed-Muller code on
inputs lying far beyond its list-decoding radius
Fourier sparsity, spectral norm, and the Log-rank conjecture
We study Boolean functions with sparse Fourier coefficients or small spectral
norm, and show their applications to the Log-rank Conjecture for XOR functions
f(x\oplus y) --- a fairly large class of functions including well studied ones
such as Equality and Hamming Distance. The rank of the communication matrix M_f
for such functions is exactly the Fourier sparsity of f. Let d be the F2-degree
of f and D^CC(f) stand for the deterministic communication complexity for
f(x\oplus y). We show that 1. D^CC(f) = O(2^{d^2/2} log^{d-2} ||\hat f||_1). In
particular, the Log-rank conjecture holds for XOR functions with constant
F2-degree. 2. D^CC(f) = O(d ||\hat f||_1) = O(\sqrt{rank(M_f)}\logrank(M_f)).
We obtain our results through a degree-reduction protocol based on a variant of
polynomial rank, and actually conjecture that its communication cost is already
\log^{O(1)}rank(M_f). The above bounds also hold for the parity decision tree
complexity of f, a measure that is no less than the communication complexity
(up to a factor of 2).
Along the way we also show several structural results about Boolean functions
with small F2-degree or small spectral norm, which could be of independent
interest. For functions f with constant F2-degree: 1) f can be written as the
summation of quasi-polynomially many indicator functions of subspaces with
\pm-signs, improving the previous doubly exponential upper bound by Green and
Sanders; 2) being sparse in Fourier domain is polynomially equivalent to having
a small parity decision tree complexity; 3) f depends only on polylog||\hat
f||_1 linear functions of input variables. For functions f with small spectral
norm: 1) there is an affine subspace with co-dimension O(||\hat f||_1) on which
f is a constant; 2) there is a parity decision tree with depth O(||\hat f||_1
log ||\hat f||_0).Comment: v2: Corollary 31 of v1 removed because of a bug in the proof. (Other
results not affected.
Why and When Can Deep -- but Not Shallow -- Networks Avoid the Curse of Dimensionality: a Review
The paper characterizes classes of functions for which deep learning can be
exponentially better than shallow learning. Deep convolutional networks are a
special case of these conditions, though weight sharing is not the main reason
for their exponential advantage
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