348 research outputs found
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
Euclidean distance geometry and applications
Euclidean distance geometry is the study of Euclidean geometry based on the
concept of distance. This is useful in several applications where the input
data consists of an incomplete set of distances, and the output is a set of
points in Euclidean space that realizes the given distances. We survey some of
the theory of Euclidean distance geometry and some of the most important
applications: molecular conformation, localization of sensor networks and
statics.Comment: 64 pages, 21 figure
Sum-of-squares proofs and the quest toward optimal algorithms
In order to obtain the best-known guarantees, algorithms are traditionally
tailored to the particular problem we want to solve. Two recent developments,
the Unique Games Conjecture (UGC) and the Sum-of-Squares (SOS) method,
surprisingly suggest that this tailoring is not necessary and that a single
efficient algorithm could achieve best possible guarantees for a wide range of
different problems.
The Unique Games Conjecture (UGC) is a tantalizing conjecture in
computational complexity, which, if true, will shed light on the complexity of
a great many problems. In particular this conjecture predicts that a single
concrete algorithm provides optimal guarantees among all efficient algorithms
for a large class of computational problems.
The Sum-of-Squares (SOS) method is a general approach for solving systems of
polynomial constraints. This approach is studied in several scientific
disciplines, including real algebraic geometry, proof complexity, control
theory, and mathematical programming, and has found applications in fields as
diverse as quantum information theory, formal verification, game theory and
many others.
We survey some connections that were recently uncovered between the Unique
Games Conjecture and the Sum-of-Squares method. In particular, we discuss new
tools to rigorously bound the running time of the SOS method for obtaining
approximate solutions to hard optimization problems, and how these tools give
the potential for the sum-of-squares method to provide new guarantees for many
problems of interest, and possibly to even refute the UGC.Comment: Survey. To appear in proceedings of ICM 201
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