2,300 research outputs found
Quantum machine learning: a classical perspective
Recently, increased computational power and data availability, as well as
algorithmic advances, have led machine learning techniques to impressive
results in regression, classification, data-generation and reinforcement
learning tasks. Despite these successes, the proximity to the physical limits
of chip fabrication alongside the increasing size of datasets are motivating a
growing number of researchers to explore the possibility of harnessing the
power of quantum computation to speed-up classical machine learning algorithms.
Here we review the literature in quantum machine learning and discuss
perspectives for a mixed readership of classical machine learning and quantum
computation experts. Particular emphasis will be placed on clarifying the
limitations of quantum algorithms, how they compare with their best classical
counterparts and why quantum resources are expected to provide advantages for
learning problems. Learning in the presence of noise and certain
computationally hard problems in machine learning are identified as promising
directions for the field. Practical questions, like how to upload classical
data into quantum form, will also be addressed.Comment: v3 33 pages; typos corrected and references adde
Reasoning about Knowledge and Strategies under Hierarchical Information
Two distinct semantics have been considered for knowledge in the context of
strategic reasoning, depending on whether players know each other's strategy or
not. The problem of distributed synthesis for epistemic temporal specifications
is known to be undecidable for the latter semantics, already on systems with
hierarchical information. However, for the other, uninformed semantics, the
problem is decidable on such systems. In this work we generalise this result by
introducing an epistemic extension of Strategy Logic with imperfect
information. The semantics of knowledge operators is uninformed, and captures
agents that can change observation power when they change strategies. We solve
the model-checking problem on a class of "hierarchical instances", which
provides a solution to a vast class of strategic problems with epistemic
temporal specifications on hierarchical systems, such as distributed synthesis
or rational synthesis
The Most Influential Paper Gerard Salton Never Wrote
Gerard Salton is often credited with developing the vector space model
(VSM) for information retrieval (IR). Citations to Salton give the impression
that the VSM must have been articulated as an IR model sometime between
1970 and 1975. However, the VSM as it is understood today evolved over a
longer time period than is usually acknowledged, and an articulation of the
model and its assumptions did not appear in print until several years after
those assumptions had been criticized and alternative models proposed. An
often cited overview paper titled ???A Vector Space Model for Information
Retrieval??? (alleged to have been published in 1975) does not exist, and
citations to it represent a confusion of two 1975 articles, neither of which
were overviews of the VSM as a model of information retrieval. Until the
late 1970s, Salton did not present vector spaces as models of IR generally
but rather as models of specifi c computations. Citations to the phantom
paper refl ect an apparently widely held misconception that the operational
features and explanatory devices now associated with the VSM must have
been introduced at the same time it was fi rst proposed as an IR model.published or submitted for publicatio
Low Rank Matrix-Valued Chernoff Bounds and Approximate Matrix Multiplication
In this paper we develop algorithms for approximating matrix multiplication
with respect to the spectral norm. Let A\in{\RR^{n\times m}} and B\in\RR^{n
\times p} be two matrices and \eps>0. We approximate the product A^\top B using
two down-sampled sketches, \tilde{A}\in\RR^{t\times m} and
\tilde{B}\in\RR^{t\times p}, where t\ll n such that \norm{\tilde{A}^\top
\tilde{B} - A^\top B} \leq \eps \norm{A}\norm{B} with high probability. We use
two different sampling procedures for constructing \tilde{A} and \tilde{B}; one
of them is done by i.i.d. non-uniform sampling rows from A and B and the other
is done by taking random linear combinations of their rows. We prove bounds
that depend only on the intrinsic dimensionality of A and B, that is their rank
and their stable rank; namely the squared ratio between their Frobenius and
operator norm. For achieving bounds that depend on rank we employ standard
tools from high-dimensional geometry such as concentration of measure arguments
combined with elaborate \eps-net constructions. For bounds that depend on the
smaller parameter of stable rank this technology itself seems weak. However, we
show that in combination with a simple truncation argument is amenable to
provide such bounds. To handle similar bounds for row sampling, we develop a
novel matrix-valued Chernoff bound inequality which we call low rank
matrix-valued Chernoff bound. Thanks to this inequality, we are able to give
bounds that depend only on the stable rank of the input matrices...Comment: 15 pages, To appear in 22nd ACM-SIAM Symposium on Discrete Algorithms
(SODA 2011
Strongly polynomial algorithm for a class of minimum-cost flow problems with separable convex objectives
A well-studied nonlinear extension of the minimum-cost flow problem is to
minimize the objective over feasible flows ,
where on every arc of the network, is a convex function. We give
a strongly polynomial algorithm for the case when all 's are convex
quadratic functions, settling an open problem raised e.g. by Hochbaum [1994].
We also give strongly polynomial algorithms for computing market equilibria in
Fisher markets with linear utilities and with spending constraint utilities,
that can be formulated in this framework (see Shmyrev [2009], Devanur et al.
[2011]). For the latter class this resolves an open question raised by Vazirani
[2010]. The running time is for quadratic costs,
for Fisher's markets with linear utilities and
for spending constraint utilities.
All these algorithms are presented in a common framework that addresses the
general problem setting. Whereas it is impossible to give a strongly polynomial
algorithm for the general problem even in an approximate sense (see Hochbaum
[1994]), we show that assuming the existence of certain black-box oracles, one
can give an algorithm using a strongly polynomial number of arithmetic
operations and oracle calls only. The particular algorithms can be derived by
implementing these oracles in the respective settings
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