23,810 research outputs found
Optimality of Geometric Local Search
International audienceUp until a decade ago, the algorithmic status of several basic NP-complete problems in geometric combinatorial optimisation was unresolved. This included the existence of polynomial-time approximation schemes (PTASs) for hitting set, set cover, dominating set, independent set, and other problems for some basic geometric objects. These past nine years have seen the resolution of all these problemsâinterestingly, with the same algorithm: local search. In fact, it was shown that for many of these problems, local search with radius λ gives a (1 + O(λ â 1 2))-approximation with running time n O(λ). Setting λ = Î(epsilon^{â2}) yields a PTAS with a running time of n^O(epsilon^{â2}). On the other hand, hardness results suggest that there do not exist PTASs for these problems with running time poly(n)·f () for any arbitrary f. Thus the main question left open in previous work is in improving the exponent of n to o(epsilon^{â2}). We show that in fact the approximation guarantee of local search cannot be improved for any of these problems. The key ingredient, of independent interest, is a new lower bound on locally expanding planar graphs, which is then used to show the impossibility results. Our construction extends to other graph families with small separators. Acknowledgements We thank the referees for several helpful comments
A Proximal-Gradient Homotopy Method for the Sparse Least-Squares Problem
We consider solving the -regularized least-squares (-LS)
problem in the context of sparse recovery, for applications such as compressed
sensing. The standard proximal gradient method, also known as iterative
soft-thresholding when applied to this problem, has low computational cost per
iteration but a rather slow convergence rate. Nevertheless, when the solution
is sparse, it often exhibits fast linear convergence in the final stage. We
exploit the local linear convergence using a homotopy continuation strategy,
i.e., we solve the -LS problem for a sequence of decreasing values of
the regularization parameter, and use an approximate solution at the end of
each stage to warm start the next stage. Although similar strategies have been
studied in the literature, there have been no theoretical analysis of their
global iteration complexity. This paper shows that under suitable assumptions
for sparse recovery, the proposed homotopy strategy ensures that all iterates
along the homotopy solution path are sparse. Therefore the objective function
is effectively strongly convex along the solution path, and geometric
convergence at each stage can be established. As a result, the overall
iteration complexity of our method is for finding an
-optimal solution, which can be interpreted as global geometric rate
of convergence. We also present empirical results to support our theoretical
analysis
On an Irreducible Theory of Complex Systems
In the paper we present results to develop an irreducible theory of complex
systems in terms of self-organization processes of prime integer relations.
Based on the integers and controlled by arithmetic only the self-organization
processes can describe complex systems by information not requiring further
explanations. Important properties of the description are revealed. It points
to a special type of correlations that do not depend on the distances between
parts, local times and physical signals and thus proposes a perspective on
quantum entanglement. Through a concept of structural complexity the
description also computationally suggests the possibility of a general
optimality condition of complex systems. The computational experiments indicate
that the performance of a complex system may behave as a concave function of
the structural complexity. A connection between the optimality condition and
the majorization principle in quantum algorithms is identified. A global
symmetry of complex systems belonging to the system as a whole, but not
necessarily applying to its embedded parts is presented. As arithmetic fully
determines the breaking of the global symmetry, there is no further need to
explain why the resulting gauge forces exist the way they do and not even
slightly different.Comment: 8 pages, 3 figures, typos are corrected, some changes and additions
are mad
Every Local Minimum Value is the Global Minimum Value of Induced Model in Non-convex Machine Learning
For nonconvex optimization in machine learning, this article proves that
every local minimum achieves the globally optimal value of the perturbable
gradient basis model at any differentiable point. As a result, nonconvex
machine learning is theoretically as supported as convex machine learning with
a handcrafted basis in terms of the loss at differentiable local minima, except
in the case when a preference is given to the handcrafted basis over the
perturbable gradient basis. The proofs of these results are derived under mild
assumptions. Accordingly, the proven results are directly applicable to many
machine learning models, including practical deep neural networks, without any
modification of practical methods. Furthermore, as special cases of our general
results, this article improves or complements several state-of-the-art
theoretical results on deep neural networks, deep residual networks, and
overparameterized deep neural networks with a unified proof technique and novel
geometric insights. A special case of our results also contributes to the
theoretical foundation of representation learning.Comment: Neural computation, MIT pres
A Sparse Multi-Scale Algorithm for Dense Optimal Transport
Discrete optimal transport solvers do not scale well on dense large problems
since they do not explicitly exploit the geometric structure of the cost
function. In analogy to continuous optimal transport we provide a framework to
verify global optimality of a discrete transport plan locally. This allows
construction of an algorithm to solve large dense problems by considering a
sequence of sparse problems instead. The algorithm lends itself to being
combined with a hierarchical multi-scale scheme. Any existing discrete solver
can be used as internal black-box.Several cost functions, including the noisy
squared Euclidean distance, are explicitly detailed. We observe a significant
reduction of run-time and memory requirements.Comment: Published "online first" in Journal of Mathematical Imaging and
Vision, see DO
On the matrix square root via geometric optimization
This paper is triggered by the preprint "\emph{Computing Matrix Squareroot
via Non Convex Local Search}" by Jain et al.
(\textit{\textcolor{blue}{arXiv:1507.05854}}), which analyzes gradient-descent
for computing the square root of a positive definite matrix. Contrary to claims
of~\citet{jain2015}, our experiments reveal that Newton-like methods compute
matrix square roots rapidly and reliably, even for highly ill-conditioned
matrices and without requiring commutativity. We observe that gradient-descent
converges very slowly primarily due to tiny step-sizes and ill-conditioning. We
derive an alternative first-order method based on geodesic convexity: our
method admits a transparent convergence analysis ( page), attains linear
rate, and displays reliable convergence even for rank deficient problems.
Though superior to gradient-descent, ultimately our method is also outperformed
by a well-known scaled Newton method. Nevertheless, the primary value of our
work is its conceptual value: it shows that for deriving gradient based methods
for the matrix square root, \emph{the manifold geometric view of positive
definite matrices can be much more advantageous than the Euclidean view}.Comment: 8 pages, 12 plots, this version contains several more references and
more words about the rank-deficient cas
The Unreasonable Success of Local Search: Geometric Optimization
What is the effectiveness of local search algorithms for geometric problems
in the plane? We prove that local search with neighborhoods of magnitude
is an approximation scheme for the following problems in the
Euclidian plane: TSP with random inputs, Steiner tree with random inputs,
facility location (with worst case inputs), and bicriteria -median (also
with worst case inputs). The randomness assumption is necessary for TSP
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