429 research outputs found
Noise-Tolerant Learning, the Parity Problem, and the Statistical Query Model
We describe a slightly sub-exponential time algorithm for learning parity
functions in the presence of random classification noise. This results in a
polynomial-time algorithm for the case of parity functions that depend on only
the first O(log n log log n) bits of input. This is the first known instance of
an efficient noise-tolerant algorithm for a concept class that is provably not
learnable in the Statistical Query model of Kearns. Thus, we demonstrate that
the set of problems learnable in the statistical query model is a strict subset
of those problems learnable in the presence of noise in the PAC model.
In coding-theory terms, what we give is a poly(n)-time algorithm for decoding
linear k by n codes in the presence of random noise for the case of k = c log n
loglog n for some c > 0. (The case of k = O(log n) is trivial since one can
just individually check each of the 2^k possible messages and choose the one
that yields the closest codeword.)
A natural extension of the statistical query model is to allow queries about
statistical properties that involve t-tuples of examples (as opposed to single
examples). The second result of this paper is to show that any class of
functions learnable (strongly or weakly) with t-wise queries for t = O(log n)
is also weakly learnable with standard unary queries. Hence this natural
extension to the statistical query model does not increase the set of weakly
learnable functions
Partition theory: A very simple illustration
We illustrate the main features of a recently proposed method based on
ensemble density functional theory to divide rigorously a complex molecular
system into its parts [M.H. Cohen and A. Wasserman, J. Phys. Chem. A 111, 2229
(2007)]. The illustrative system is an analog of the hydrogen molecule for
which analytic expressions for the densities of the parts (hydrogen "atoms")
are found along with the "reactivity potential" that enters the theory. While
previous formulations of Chemical Reactivity Theory lead to zero, or undefined,
values for the chemical hardness of the isolated parts, we demonstrate they can
acquire a finite and positive hardness within the present formulation.Comment: 8 pages, 8 figure
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