Polynomial-time Tensor Decompositions with Sum-of-Squares

We give new algorithms based on the sum-of-squares method for tensor decomposition. Our results improve the best known running times from quasi-polynomial to polynomial for several problems, including decomposing random overcomplete 3-tensors and learning overcomplete dictionaries with constant relative sparsity. We also give the first robust analysis for decomposing overcomplete 4-tensors in the smoothed analysis model. A key ingredient of our analysis is to establish small spectral gaps in moment matrices derived from solutions to sum-of-squares relaxations. To enable this analysis we augment sum-of-squares relaxations with spectral analogs of maximum entropy constraints.Comment: to appear in FOCS 201

China's Changing Energy Intensity Trend: A Decomposition Analysis

China experienced a dramatic decline in energy intensity from the onset of economic reform in the late 1970s until 2000, but since then rate of decline slowed and energy intensity actually increased in 2003. Most previous studies found that most of the decline was due to technological change, but disagreed on the role of structural change. To the best of our knowledge, no decomposition study has investigated the role of inter-fuel substitution in the decline in energy intensity or the causes of the rise in energy intensity since 2000. In this paper, we use logarithmic mean Divisia index (LMDI) techniques to decompose changes in energy intensity in the period 1980-2003. We find that: (1) technological change is confirmed as the dominant contributor to the decline in energy intensity; (2) structural change at the industry and sector (sub-industry) level actually increased energy intensity over the period of 1980-2003, although the structural change at the industry level was very different in the 1980s and in the post 1990 period; (3) structural change involving shifts of production between sub-sectors, however, decreased overall energy intensity; (4) the increase in energy intensity since 2000 is explained by negative technological progress; (5) inter-fuel substitution is found to contribute little to the changes in energy intensity.

Some combinatorial arrays related to the Lotka-Volterra system

The purpose of this paper is to investigate the connection between the Lotka-Volterra system and combinatorics. We study several context-free grammars associated with the Lotka-Volterra system. Some combinatorial arrays, involving the Stirling numbers of the second kind and Eulerian numbers, are generated by these context-free grammars. In particular, we present grammatical characterization of some statistics on cyclically ordered partitions.Comment: 15 page

Restricted Stirling permutations

In this paper, we study the generating functions for the number of pattern restricted Stirling permutations with a given number of plateaus, descents and ascents. Properties of the generating functions, including symmetric properties and explicit formulas are studied. Combinatorial explanations are given for some equidistributions.Comment: 14 pages in Taiwanese Journal of Mathematics, 201

On Policies for Single-leg Revenue Management with Limited Demand Information

In this paper we study the single-item revenue management problem, with no information given about the demand trajectory over time. When the item is sold through accepting/rejecting different fare classes, Ball and Queyranne (2009) have established the tight competitive ratio for this problem using booking limit policies, which raise the acceptance threshold as the remaining inventory dwindles. However, when the item is sold through dynamic pricing instead, there is the additional challenge that offering a low price may entice high-paying customers to substitute down. We show that despite this challenge, the same competitive ratio can still be achieved using a randomized dynamic pricing policy. Our policy incorporates the price-skimming technique from Eren and Maglaras (2010), but importantly we show how the randomized price distribution should be stochastically-increased as the remaining inventory dwindles. A key technical ingredient in our policy is a new "valuation tracking" subroutine, which tracks the possible values for the optimum, and follows the most "inventory-conservative" control which maintains the desired competitive ratio. Finally, we demonstrate the empirical effectiveness of our policy in simulations, where its average-case performance surpasses all naive modifications of the existing policies