8,341 research outputs found
Statistical mechanics of lossy data compression using a non-monotonic perceptron
The performance of a lossy data compression scheme for uniformly biased
Boolean messages is investigated via methods of statistical mechanics. Inspired
by a formal similarity to the storage capacity problem in the research of
neural networks, we utilize a perceptron of which the transfer function is
appropriately designed in order to compress and decode the messages. Employing
the replica method, we analytically show that our scheme can achieve the
optimal performance known in the framework of lossy compression in most cases
when the code length becomes infinity. The validity of the obtained results is
numerically confirmed.Comment: 9 pages, 5 figures, Physical Review
Schumacher's quantum data compression as a quantum computation
An explicit algorithm for performing Schumacher's noiseless compression of
quantum bits is given. This algorithm is based on a combinatorial expression
for a particular bijection among binary strings. The algorithm, which adheres
to the rules of reversible programming, is expressed in a high-level pseudocode
language. It is implemented using two- and three-bit primitive
reversible operations, where is the length of the qubit strings to be
compressed. Also, the algorithm makes use of auxiliary qubits; however,
space-saving techniques based on those proposed by Bennett are developed which
reduce this workspace to while increasing the running time by
less than a factor of two.Comment: 37 pages, no figure
Online Learning of k-CNF Boolean Functions
This paper revisits the problem of learning a k-CNF Boolean function from
examples in the context of online learning under the logarithmic loss. In doing
so, we give a Bayesian interpretation to one of Valiant's celebrated PAC
learning algorithms, which we then build upon to derive two efficient, online,
probabilistic, supervised learning algorithms for predicting the output of an
unknown k-CNF Boolean function. We analyze the loss of our methods, and show
that the cumulative log-loss can be upper bounded, ignoring logarithmic
factors, by a polynomial function of the size of each example.Comment: 20 LaTeX pages. 2 Algorithms. Some Theorem
Algorithms for Provisioning Queries and Analytics
Provisioning is a technique for avoiding repeated expensive computations in
what-if analysis. Given a query, an analyst formulates hypotheticals, each
retaining some of the tuples of a database instance, possibly overlapping, and
she wishes to answer the query under scenarios, where a scenario is defined by
a subset of the hypotheticals that are "turned on". We say that a query admits
compact provisioning if given any database instance and any hypotheticals,
one can create a poly-size (in ) sketch that can then be used to answer the
query under any of the possible scenarios without accessing the
original instance.
In this paper, we focus on provisioning complex queries that combine
relational algebra (the logical component), grouping, and statistics/analytics
(the numerical component). We first show that queries that compute quantiles or
linear regression (as well as simpler queries that compute count and
sum/average of positive values) can be compactly provisioned to provide
(multiplicative) approximate answers to an arbitrary precision. In contrast,
exact provisioning for each of these statistics requires the sketch size to be
exponential in . We then establish that for any complex query whose logical
component is a positive relational algebra query, as long as the numerical
component can be compactly provisioned, the complex query itself can be
compactly provisioned. On the other hand, introducing negation or recursion in
the logical component again requires the sketch size to be exponential in .
While our positive results use algorithms that do not access the original
instance after a scenario is known, we prove our lower bounds even for the case
when, knowing the scenario, limited access to the instance is allowed
Chain Reduction for Binary and Zero-Suppressed Decision Diagrams
Chain reduction enables reduced ordered binary decision diagrams (BDDs) and
zero-suppressed binary decision diagrams (ZDDs) to each take advantage of the
others' ability to symbolically represent Boolean functions in compact form.
For any Boolean function, its chain-reduced ZDD (CZDD) representation will be
no larger than its ZDD representation, and at most twice the size of its BDD
representation. The chain-reduced BDD (CBDD) of a function will be no larger
than its BDD representation, and at most three times the size of its CZDD
representation. Extensions to the standard algorithms for operating on BDDs and
ZDDs enable them to operate on the chain-reduced versions. Experimental
evaluations on representative benchmarks for encoding word lists, solving
combinatorial problems, and operating on digital circuits indicate that chain
reduction can provide significant benefits in terms of both memory and
execution time
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