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

SqueezeSeg: Convolutional Neural Nets with Recurrent CRF for Real-Time Road-Object Segmentation from 3D LiDAR Point Cloud

In this paper, we address semantic segmentation of road-objects from 3D LiDAR point clouds. In particular, we wish to detect and categorize instances of interest, such as cars, pedestrians and cyclists. We formulate this problem as a point- wise classification problem, and propose an end-to-end pipeline called SqueezeSeg based on convolutional neural networks (CNN): the CNN takes a transformed LiDAR point cloud as input and directly outputs a point-wise label map, which is then refined by a conditional random field (CRF) implemented as a recurrent layer. Instance-level labels are then obtained by conventional clustering algorithms. Our CNN model is trained on LiDAR point clouds from the KITTI dataset, and our point-wise segmentation labels are derived from 3D bounding boxes from KITTI. To obtain extra training data, we built a LiDAR simulator into Grand Theft Auto V (GTA-V), a popular video game, to synthesize large amounts of realistic training data. Our experiments show that SqueezeSeg achieves high accuracy with astonishingly fast and stable runtime (8.7 ms per frame), highly desirable for autonomous driving applications. Furthermore, additionally training on synthesized data boosts validation accuracy on real-world data. Our source code and synthesized data will be open-sourced

Enhancement of Dark Matter Annihilation via Breit-Wigner Resonance

The Breit-Wigner enhancement of the thermally averaged annihilation cross section is shown to provide a large boost factor when the dark matter annihilation process nears a narrow resonance. We explicitly demonstrate the evolution behavior of the Breit-Wigner enhanced as the function of universe temperature for both the physical and unphysical pole cases. It is found that both of the cases can lead an enough large boost factor to explain the recent PAMELA, ATIC and PPB-BETS anomalies. We also calculate the coupling of annihilation process, which is useful for an appropriate model building to give the desired dark matter relic density.Comment: 4 pages, 4 figures, references added, accepted for publication in Physical Review

Direct detection and solar capture of dark matter with momentum and velocity dependent elastic scattering

We explore the momentum and velocity dependent elastic scattering between the dark matter (DM) particles and the nuclei in detectors and the Sun. In terms of the non-relativistic effective theory, we phenomenologically discuss ten kinds of momentum and velocity dependent DM-nucleus interactions and recalculate the corresponding upper limits on the spin-independent DM-nucleon scattering cross section from the current direct detection experiments. The DM solar capture rate is calculated for each interaction. Our numerical results show that the momentum and velocity dependent cases can give larger solar capture rate than the usual contact interaction case for almost the whole parameter space. On the other hand, we deduce the Super-Kamiokande's constraints on the solar capture rate for eight typical DM annihilation channels. In contrast to the usual contact interaction, the Super-Kamiokande and IceCube experiments can give more stringent limits on the DM-nucleon elastic scattering cross section than the current direct detection experiments for several momentum and velocity dependent DM-nucleus interactions. In addition, we investigate the mediator mass's effect on the DM elastic scattering cross section and solar capture rate.Comment: 18 pages, 4 figures, 2 tables. minor changes and a reference added, published in Nuclear Physics