Decision Trees, Protocols, and the Fourier Entropy-Influence Conjecture

Given $f:\{-1, 1\}^n \rightarrow \{-1, 1\}$, define the \emph{spectral distribution} of $f$ to be the distribution on subsets of $[n]$ in which the set $S$ is sampled with probability $\widehat{f}(S)^2$. Then the Fourier Entropy-Influence (FEI) conjecture of Friedgut and Kalai (1996) states that there is some absolute constant $C$ such that $\operatorname{H}[\widehat{f}^2] \leq C\cdot\operatorname{Inf}[f]$. Here, $\operatorname{H}[\widehat{f}^2]$ denotes the Shannon entropy of $f$'s spectral distribution, and $\operatorname{Inf}[f]$ is the total influence of $f$. This conjecture is one of the major open problems in the analysis of Boolean functions, and settling it would have several interesting consequences. Previous results on the FEI conjecture have been largely through direct calculation. In this paper we study a natural interpretation of the conjecture, which states that there exists a communication protocol which, given subset $S$ of $[n]$ distributed as $\widehat{f}^2$, can communicate the value of $S$ using at most $C\cdot\operatorname{Inf}[f]$ bits in expectation. Using this interpretation, we are able show the following results: 1. First, if $f$ is computable by a read-$k$ decision tree, then $\operatorname{H}[\widehat{f}^2] \leq 9k\cdot \operatorname{Inf}[f]$. 2. Next, if $f$ has $\operatorname{Inf}[f] \geq 1$ and is computable by a decision tree with expected depth $d$, then $\operatorname{H}[\widehat{f}^2] \leq 12d\cdot \operatorname{Inf}[f]$. 3. Finally, we give a new proof of the main theorem of O'Donnell and Tan (ICALP 2013), i.e. that their FEI$^+$ conjecture composes. In addition, we show that natural improvements to our decision tree results would be sufficient to prove the FEI conjecture in its entirety. We believe that our methods give more illuminating proofs than previous results about the FEI conjecture

Hardness of robust graph isomorphism, Lasserre gaps, and asymmetry of random graphs

Building on work of Cai, F\"urer, and Immerman \cite{CFI92}, we show two hardness results for the Graph Isomorphism problem. First, we show that there are pairs of nonisomorphic $n$-vertex graphs $G$ and $H$ such that any sum-of-squares (SOS) proof of nonisomorphism requires degree $\Omega(n)$. In other words, we show an $\Omega(n)$-round integrality gap for the Lasserre SDP relaxation. In fact, we show this for pairs $G$ and $H$ which are not even $(1-10^{-14})$-isomorphic. (Here we say that two $n$-vertex, $m$-edge graphs $G$ and $H$ are $\alpha$-isomorphic if there is a bijection between their vertices which preserves at least $\alpha m$ edges.) Our second result is that under the {\sc R3XOR} Hypothesis \cite{Fei02} (and also any of a class of hypotheses which generalize the {\sc R3XOR} Hypothesis), the \emph{robust} Graph Isomorphism problem is hard. I.e.\ for every $\epsilon > 0$, there is no efficient algorithm which can distinguish graph pairs which are $(1-\epsilon)$-isomorphic from pairs which are not even $(1-\epsilon_0)$-isomorphic for some universal constant $\epsilon_0$. Along the way we prove a robust asymmetry result for random graphs and hypergraphs which may be of independent interest

Electron Transport Chain Is Biochemically Linked to Pilus Assembly Required for Polymicrobial Interactions and Biofilm Formation in the Gram-Positive Actinobacterium Actinomyces oris.

The Gram-positive actinobacteria Actinomyces spp. are key colonizers in the development of oral biofilms due to the inherent ability of Actinomyces to adhere to receptor polysaccharides on the surface of oral streptococci and host cells. This receptor-dependent bacterial interaction, or coaggregation, requires a unique sortase-catalyzed pilus consisting of the pilus shaft FimA and the coaggregation factor CafA forming the pilus tip. While the essential role of the sortase machine SrtC2 in pilus assembly, biofilm formation, and coaggregation has been established, little is known about trans-acting factors contributing to these processes. We report here a large-scale Tn5 transposon screen for mutants defective in Actinomyces oris coaggregation with Streptococcus oralis We obtained 33 independent clones, 13 of which completely failed to aggregate with S. oralis, and the remainder of which exhibited a range of phenotypes from severely to weakly defective coaggregation. The former had Tn5 insertions in fimA, cafA, or srtC2, as expected; the latter were mapped to genes coding for uncharacterized proteins and various nuo genes encoding the NADH dehydrogenase subunits. Electron microscopy and biochemical analyses of mutants with nonpolar deletions of nuo genes and ubiE, a menaquinone C-methyltransferase-encoding gene downstream of the nuo locus, confirmed the pilus and coaggregation defects. Both nuoA and ubiE mutants were defective in oxidation of MdbA, the major oxidoreductase required for oxidative folding of pilus proteins. Furthermore, supplementation of the ubiE mutant with exogenous menaquinone-4 rescued the cell growth and pilus defects. Altogether, we propose that the A. oris electron transport chain is biochemically linked to pilus assembly via oxidative protein folding.IMPORTANCE The Gram-positive actinobacterium A. oris expresses adhesive pili, or fimbriae, that are essential to biofilm formation and Actinomyces interactions with other bacteria, termed coaggregation. While the critical role of the conserved sortase machine in pilus assembly and the disulfide bond-forming catalyst MdbA in oxidative folding of pilins has been established, little is known about other trans-acting factors involved in these processes. Using a Tn5 transposon screen for mutants defective in coaggregation with Streptococcus oralis, we found that genetic disruption of the NADH dehydrogenase and menaquinone biosynthesis detrimentally alters pilus assembly. Further biochemical characterizations determined that menaquinone is important for reactivation of MdbA. This study supports the notion that the electron transport chain is biochemically linked to pilus assembly in A. oris via oxidative folding of pilin precursors

Improving Person Re-identification by Attribute and Identity Learning

Person re-identification (re-ID) and attribute recognition share a common target at learning pedestrian descriptions. Their difference consists in the granularity. Most existing re-ID methods only take identity labels of pedestrians into consideration. However, we find the attributes, containing detailed local descriptions, are beneficial in allowing the re-ID model to learn more discriminative feature representations. In this paper, based on the complementarity of attribute labels and ID labels, we propose an attribute-person recognition (APR) network, a multi-task network which learns a re-ID embedding and at the same time predicts pedestrian attributes. We manually annotate attribute labels for two large-scale re-ID datasets, and systematically investigate how person re-ID and attribute recognition benefit from each other. In addition, we re-weight the attribute predictions considering the dependencies and correlations among the attributes. The experimental results on two large-scale re-ID benchmarks demonstrate that by learning a more discriminative representation, APR achieves competitive re-ID performance compared with the state-of-the-art methods. We use APR to speed up the retrieval process by ten times with a minor accuracy drop of 2.92% on Market-1501. Besides, we also apply APR on the attribute recognition task and demonstrate improvement over the baselines.Comment: Accepted to Pattern Recognition (PR