Spectral Sparsification and Regret Minimization Beyond Matrix Multiplicative Updates

In this paper, we provide a novel construction of the linear-sized spectral sparsifiers of Batson, Spielman and Srivastava [BSS14]. While previous constructions required $\Omega(n^4)$ running time [BSS14, Zou12], our sparsification routine can be implemented in almost-quadratic running time $O(n^{2+\varepsilon})$. The fundamental conceptual novelty of our work is the leveraging of a strong connection between sparsification and a regret minimization problem over density matrices. This connection was known to provide an interpretation of the randomized sparsifiers of Spielman and Srivastava [SS11] via the application of matrix multiplicative weight updates (MWU) [CHS11, Vis14]. In this paper, we explain how matrix MWU naturally arises as an instance of the Follow-the-Regularized-Leader framework and generalize this approach to yield a larger class of updates. This new class allows us to accelerate the construction of linear-sized spectral sparsifiers, and give novel insights on the motivation behind Batson, Spielman and Srivastava [BSS14]

Inapproximability of maximal strip recovery

In comparative genomic, the first step of sequence analysis is usually to decompose two or more genomes into syntenic blocks that are segments of homologous chromosomes. For the reliable recovery of syntenic blocks, noise and ambiguities in the genomic maps need to be removed first. Maximal Strip Recovery (MSR) is an optimization problem proposed by Zheng, Zhu, and Sankoff for reliably recovering syntenic blocks from genomic maps in the midst of noise and ambiguities. Given $d$ genomic maps as sequences of gene markers, the objective of \msr{d} is to find $d$ subsequences, one subsequence of each genomic map, such that the total length of syntenic blocks in these subsequences is maximized. For any constant $d \ge 2$, a polynomial-time 2d-approximation for \msr{d} was previously known. In this paper, we show that for any $d \ge 2$, \msr{d} is APX-hard, even for the most basic version of the problem in which all gene markers are distinct and appear in positive orientation in each genomic map. Moreover, we provide the first explicit lower bounds on approximating \msr{d} for all $d \ge 2$. In particular, we show that \msr{d} is NP-hard to approximate within $\Omega(d/\log d)$. From the other direction, we show that the previous 2d-approximation for \msr{d} can be optimized into a polynomial-time algorithm even if $d$ is not a constant but is part of the input. We then extend our inapproximability results to several related problems including \cmsr{d}, \gapmsr{\delta}{d}, and \gapcmsr{\delta}{d}.Comment: A preliminary version of this paper appeared in two parts in the Proceedings of the 20th International Symposium on Algorithms and Computation (ISAAC 2009) and the Proceedings of the 4th International Frontiers of Algorithmics Workshop (FAW 2010

On k-Column Sparse Packing Programs

We consider the class of packing integer programs (PIPs) that are column sparse, i.e. there is a specified upper bound k on the number of constraints that each variable appears in. We give an (ek+o(k))-approximation algorithm for k-column sparse PIPs, improving on recent results of $k^2\cdot 2^k$ and $O(k^2)$. We also show that the integrality gap of our linear programming relaxation is at least 2k-1; it is known that k-column sparse PIPs are $\Omega(k/ \log k)$-hard to approximate. We also extend our result (at the loss of a small constant factor) to the more general case of maximizing a submodular objective over k-column sparse packing constraints.Comment: 19 pages, v3: additional detail

Competitive portfolio selection using stochastic predictions

We study a portfolio selection problem where a player attempts to maximise a utility function that represents the growth rate of wealth. We show that, given some stochastic predictions of the asset prices in the next time step, a sublinear expected regret is attainable against an optimal greedy algorithm, subject to tradeoff against the \accuracy" of such predictions that learn (or improve) over time. We also study the effects of introducing transaction costs into the model

Social welfare and profit maximization from revealed preferences

Consider the seller's problem of finding optimal prices for her $n$ (divisible) goods when faced with a set of $m$ consumers, given that she can only observe their purchased bundles at posted prices, i.e., revealed preferences. We study both social welfare and profit maximization with revealed preferences. Although social welfare maximization is a seemingly non-convex optimization problem in prices, we show that (i) it can be reduced to a dual convex optimization problem in prices, and (ii) the revealed preferences can be interpreted as supergradients of the concave conjugate of valuation, with which subgradients of the dual function can be computed. We thereby obtain a simple subgradient-based algorithm for strongly concave valuations and convex cost, with query complexity $O(m^2/\epsilon^2)$, where $\epsilon$ is the additive difference between the social welfare induced by our algorithm and the optimum social welfare. We also study social welfare maximization under the online setting, specifically the random permutation model, where consumers arrive one-by-one in a random order. For the case where consumer valuations can be arbitrary continuous functions, we propose a price posting mechanism that achieves an expected social welfare up to an additive factor of $O(\sqrt{mn})$ from the maximum social welfare. Finally, for profit maximization (which may be non-convex in simple cases), we give nearly matching upper and lower bounds on the query complexity for separable valuations and cost (i.e., each good can be treated independently)

Dynamics of Transformation from Segregation to Mixed Wealth Cities

We model the dynamics of the Schelling model for agents described simply by a continuously distributed variable - wealth. Agents move to neighborhoods where their wealth is not lesser than that of some proportion of their neighbors, the threshold level. As in the case of the classic Schelling model where segregation obtains between two races, we find here that wealth-based segregation occurs and persists. However, introducing uncertainty into the decision to move - that is, with some probability, if agents are allowed to move even though the threshold level condition is contravened - we find that even for small proportions of such disallowed moves, the dynamics no longer yield segregation but instead sharply transition into a persistent mixed wealth distribution. We investigate the nature of this sharp transformation between segregated and mixed states, and find that it is because of a non-linear relationship between allowed moves and disallowed moves. For small increases in disallowed moves, there is a rapid corresponding increase in allowed moves, but this tapers off as the fraction of disallowed moves increase further and finally settles at a stable value, remaining invariant to any further increase in disallowed moves. It is the overall effect of the dynamics in the initial region (with small numbers of disallowed moves) that shifts the system away from a state of segregation rapidly to a mixed wealth state. The contravention of the tolerance condition could be interpreted as public policy interventions like minimal levels of social housing or housing benefit transfers to poorer households. Our finding therefore suggests that it might require only very limited levels of such public intervention - just sufficient to enable a small fraction of disallowed moves, because the dynamics generated by such moves could spur the transformation from a segregated to mixed equilibrium.Comment: 12 pages, 7 figure

SNP microarray-based 24 chromosome aneuploidy screening is significantly more consistent than FISH

Many studies estimate that chromosomal mosaicism within the cleavage-stage human embryo is high. However, comparison of two unique methods of aneuploidy screening of blastomeres within the same embryo has not been conducted and may indicate whether mosaicism has been overestimated due to technical inconsistency rather than the biological phenomena. The present study investigates the prevalence of chromosomal abnormality and mosaicism found with two different single cell aneuploidy screening techniques. Thirteen arrested cleavage-stage embryos were studied. Each was biopsied into individual cells (n = 160). The cells from each embryo were randomized into two groups. Those destined for FISH-based aneuploidy screening (n = 75) were fixed, one cell per slide. Cells for SNP microarray-based aneuploidy screening (n = 85) were put into individual tubes. Microarray was significantly more reliable (96%) than FISH (83%) for providing an interpretable result (P = 0.004). Markedly different results were obtained when comparing microarray and FISH results from individual embryos. Mosaicism was significantly less commonly observed by microarray (31%) than by FISH (100%) (P = 0.0005). Although FISH evaluated fewer chromosomes per cell and fewer cells per embryo, FISH still displayed significantly more unique genetic diagnoses per embryo (3.2 ± 0.2) than microarray (1.3 ± 0.2) (P < 0.0001). This is the first prospective, randomized, blinded and paired comparison between microarray and FISH-based aneuploidy screening. SNP microarray-based 24 chromosome aneuploidy screening provides more complete and consistent results than FISH. These results also suggest that FISH technology may overestimate the contribution of mitotic error to the origin of aneuploidy at the cleavage stage of human embryogenesis

The complex TIE between macrophages and angiogenesis

Macrophages are primarily known as phagocytic immune cells, but they also play a role in diverse processes, such as morphogenesis, homeostasis and regeneration. In this review, we discuss the influence of macrophages on angiogenesis, the process of new blood vessel formation from the pre-existing vasculature. Macrophages play crucial roles at each step of the angiogenic cascade, starting from new blood vessel sprouting to the remodelling of the vascular plexus and vessel maturation. Macrophages form promising targets for both pro- and anti-angiogenic treatments. However, to target macrophages, we will first need to understand the mechanisms that control the functional plasticity of macrophages during each of the steps of the angiogenic cascade. Here, we review recent insights in this topic. Special attention will be given to the TIE2-expressing macrophage (TEM), which is a subtype of highly angiogenic macrophages that is able to influence angiogenesis via the angiopoietin-TIE pathway

Auditory Enhancement and Second Language Experience in Spanish and English Weighting of Secondary Voicing Cues

The role of secondary cues in voicing categorization was investigated in three listener groups: Monolingual English (n=20) and Spanish speakers (n=20), and Spanish speakers with significant English experience (n=16). Results showed that, in all three groups, participants used onset f0 in making voicing decisions only in the positive voice onset time (VOT) range (short lag and long lag tokens), while there was no effect of onset f0 on voicing categorization within the negative VOT range (voicing lead tokens) for any of the participant groups. These results support an auditory enhancement view of perceptual cue weighting: Onset f0 serves as a secondary cue to voicing only in the positive VOT range where it is not overshadowed by the presence of pre-voicing. Moreover, results showed that Spanish learners of English gave a significantly greater weight to onset f0 in their voicing decisions than did listeners in either of the other two groups. This result supports the view that learners may overweight secondary cues to distinguish between non-native categories that are assimilated to the same native category on the basis of a primary cue