11 research outputs found
Learning Unions of -Dimensional Rectangles
We consider the problem of learning unions of rectangles over the domain
, in the uniform distribution membership query learning setting, where
both b and n are "large". We obtain poly-time algorithms for the
following classes:
- poly-way Majority of -dimensional rectangles.
- Union of poly many -dimensional rectangles.
- poly-way Majority of poly-Or of disjoint
-dimensional rectangles.
Our main algorithmic tool is an extension of Jackson's boosting- and
Fourier-based Harmonic Sieve algorithm [Jackson 1997] to the domain ,
building on work of [Akavia, Goldwasser, Safra 2003]. Other ingredients used to
obtain the results stated above are techniques from exact learning [Beimel,
Kushilevitz 1998] and ideas from recent work on learning augmented
circuits [Jackson, Klivans, Servedio 2002] and on representing Boolean
functions as thresholds of parities [Klivans, Servedio 2001].Comment: 25 pages. Some corrections. Recipient of E. M. Gold award ALT 2006.
To appear in Journal of Theoretical Computer Scienc
Algorithmic Polynomials
The approximate degree of a Boolean function is
the minimum degree of a real polynomial that approximates pointwise within
. Upper bounds on approximate degree have a variety of applications in
learning theory, differential privacy, and algorithm design in general. Nearly
all known upper bounds on approximate degree arise in an existential manner
from bounds on quantum query complexity. We develop a first-principles,
classical approach to the polynomial approximation of Boolean functions. We use
it to give the first constructive upper bounds on the approximate degree of
several fundamental problems:
- for the -element
distinctness problem;
- for the -subset sum problem;
- for any -DNF or -CNF formula;
- for the surjectivity problem.
In all cases, we obtain explicit, closed-form approximating polynomials that
are unrelated to the quantum arguments from previous work. Our first three
results match the bounds from quantum query complexity. Our fourth result
improves polynomially on the quantum query complexity of the
problem and refutes the conjecture by several experts that surjectivity has
approximate degree . In particular, we exhibit the first natural
problem with a polynomial gap between approximate degree and quantum query
complexity
Near-Optimal Lower Bounds on the Threshold Degree and Sign-Rank of AC^0
The threshold degree of a Boolean function is
the minimum degree of a real polynomial that represents in sign:
A related notion is sign-rank, defined for a
Boolean matrix as the minimum rank of a real matrix with
. Determining the maximum threshold degree
and sign-rank achievable by constant-depth circuits () is a
well-known and extensively studied open problem, with complexity-theoretic and
algorithmic applications.
We give an essentially optimal solution to this problem. For any
we construct an circuit in variables that has
threshold degree and sign-rank
improving on the previous best lower bounds of
and , respectively. Our
results subsume all previous lower bounds on the threshold degree and sign-rank
of circuits of any given depth, with a strict improvement
starting at depth . As a corollary, we also obtain near-optimal bounds on
the discrepancy, threshold weight, and threshold density of ,
strictly subsuming previous work on these quantities. Our work gives some of
the strongest lower bounds to date on the communication complexity of
.Comment: 99 page
Pattern recognition in the nucleation kinetics of non-equilibrium self-assembly
Inspired by biology’s most sophisticated computer, the brain, neural networks constitute a profound reformulation of computational principles. Analogous high-dimensional, highly interconnected computational architectures also arise within information-processing molecular systems inside living cells, such as signal transduction cascades and genetic regulatory networks. Might collective modes analogous to neural computation be found more broadly in other physical and chemical processes, even those that ostensibly play non-information-processing roles? Here we examine nucleation during self-assembly of multicomponent structures, showing that high-dimensional patterns of concentrations can be discriminated and classified in a manner similar to neural network computation. Specifically, we design a set of 917 DNA tiles that can self-assemble in three alternative ways such that competitive nucleation depends sensitively on the extent of colocalization of high-concentration tiles within the three structures. The system was trained in silico to classify a set of 18 grayscale 30 × 30 pixel images into three categories. Experimentally, fluorescence and atomic force microscopy measurements during and after a 150 hour anneal established that all trained images were correctly classified, whereas a test set of image variations probed the robustness of the results. Although slow compared to previous biochemical neural networks, our approach is compact, robust and scalable. Our findings suggest that ubiquitous physical phenomena, such as nucleation, may hold powerful information-processing capabilities when they occur within high-dimensional multicomponent systems