6,624 research outputs found
Exploring Algorithmic Limits of Matrix Rank Minimization under Affine Constraints
Many applications require recovering a matrix of minimal rank within an
affine constraint set, with matrix completion a notable special case. Because
the problem is NP-hard in general, it is common to replace the matrix rank with
the nuclear norm, which acts as a convenient convex surrogate. While elegant
theoretical conditions elucidate when this replacement is likely to be
successful, they are highly restrictive and convex algorithms fail when the
ambient rank is too high or when the constraint set is poorly structured.
Non-convex alternatives fare somewhat better when carefully tuned; however,
convergence to locally optimal solutions remains a continuing source of
failure. Against this backdrop we derive a deceptively simple and
parameter-free probabilistic PCA-like algorithm that is capable, over a wide
battery of empirical tests, of successful recovery even at the theoretical
limit where the number of measurements equal the degrees of freedom in the
unknown low-rank matrix. Somewhat surprisingly, this is possible even when the
affine constraint set is highly ill-conditioned. While proving general recovery
guarantees remains evasive for non-convex algorithms, Bayesian-inspired or
otherwise, we nonetheless show conditions whereby the underlying cost function
has a unique stationary point located at the global optimum; no existing cost
function we are aware of satisfies this same property. We conclude with a
simple computer vision application involving image rectification and a standard
collaborative filtering benchmark
Automatic Image Segmentation by Dynamic Region Merging
This paper addresses the automatic image segmentation problem in a region
merging style. With an initially over-segmented image, in which the many
regions (or super-pixels) with homogeneous color are detected, image
segmentation is performed by iteratively merging the regions according to a
statistical test. There are two essential issues in a region merging algorithm:
order of merging and the stopping criterion. In the proposed algorithm, these
two issues are solved by a novel predicate, which is defined by the sequential
probability ratio test (SPRT) and the maximum likelihood criterion. Starting
from an over-segmented image, neighboring regions are progressively merged if
there is an evidence for merging according to this predicate. We show that the
merging order follows the principle of dynamic programming. This formulates
image segmentation as an inference problem, where the final segmentation is
established based on the observed image. We also prove that the produced
segmentation satisfies certain global properties. In addition, a faster
algorithm is developed to accelerate the region merging process, which
maintains a nearest neighbor graph in each iteration. Experiments on real
natural images are conducted to demonstrate the performance of the proposed
dynamic region merging algorithm.Comment: 28 pages. This paper is under review in IEEE TI
Evaluation of Corporate Sustainability
As a consequence of an increasing demand in sustainable development for business organizations, the evaluation of corporate sustainability has become a topic intensively focused by academic researchers and business practitioners. Several techniques in the context of multiple criteria decision analysis (MCDA) have been suggested to facilitate the evaluation and the analysis of sustainability performance. However, due to the complexity of evaluation, such as a compilation of quantitative and qualitative measures, interrelationships among various sustainability criteria, the assessor’s hesitation in scoring, or incomplete information, simple techniques may not be able to generate reliable results which can reflect the overall sustainability performance of a company. This paper proposes a series of mathematical formulations based upon the evidential reasoning (ER) approach which can be used to aggregate results from qualitative judgments with quantitative measurements under various types of complex and uncertain situations. The evaluation of corporate sustainability through the ER model is demonstrated using actual data generated from three sugar manufacturing companies in Thailand. The proposed model facilitates managers in analysing the performance and identifying improvement plans and goals. It also simplifies decision making related to sustainable development initiatives. The model can be generalized to a wider area of performance assessment, as well as to any cases of multiple criteria analysis
Three `species' of Schr\"odinger cat states in an infinite-range spin model
We explore a transverse-field Ising model that exhibits both spontaneous
symmetry-breaking and eigenstate thermalization. Within its ferromagnetic
phase, the exact eigenstates of the Hamiltonian of any large but finite-sized
system are all Schr\"odinger cat states: superpositions of states with `up' and
`down' spontaneous magnetization. This model exhibits two dynamical phase
transitions {\it within} its ferromagnetic phase: In the lowest-temperature
phase the magnetization can macroscopically oscillate between up and down. The
relaxation of the magnetization is always overdamped in the remainder of the
ferromagnetic phase, which is divided in to phases where the system thermally
activates itself {\it over} the barrier between the up and down states, and
where it quantum tunnels.Comment: 7 pages, added numerical result
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