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

Mining Measured Information from Text

We present an approach to extract measured information from text (e.g., a 1370 degrees C melting point, a BMI greater than 29.9 kg/m^2 ). Such extractions are critically important across a wide range of domains - especially those involving search and exploration of scientific and technical documents. We first propose a rule-based entity extractor to mine measured quantities (i.e., a numeric value paired with a measurement unit), which supports a vast and comprehensive set of both common and obscure measurement units. Our method is highly robust and can correctly recover valid measured quantities even when significant errors are introduced through the process of converting document formats like PDF to plain text. Next, we describe an approach to extracting the properties being measured (e.g., the property "pixel pitch" in the phrase "a pixel pitch as high as 352 {\mu}m"). Finally, we present MQSearch: the realization of a search engine with full support for measured information.Comment: 4 pages; 38th International ACM SIGIR Conference on Research and Development in Information Retrieval (SIGIR '15

Characterisation and optimisation of PECVD SiNx as an antireflection coating and passivation layer for silicon solar cells

In this work, we investigate how the film properties of silicon nitride (SiNx) depend on its deposition conditions when formed by plasma enhanced chemical vapour deposition (PECVD). The examination is conducted with a Roth & Rau AK400 PECVD reactor, where the varied parameters are deposition temperature, pressure, gas flow ratio, total gas flow, microwave plasma power and radio-frequency bias voltage. The films are evaluated by Fourier transform infrared spectroscopy to determine structural properties, by spectrophotometry to determine optical properties, and by capacitance–voltage and photoconductance measurements to determine electronic properties. After reporting on the dependence of SiNx properties on deposition parameters, we determine the optimized deposition conditions that attain low absorption and low recombination. On the basis of SiNx growth models proposed in the literature and of our experimental results, we discuss how each process parameter affects the deposition rate and chemical bond density. We then focus on the effective surface recombination velocity S eff, which is of primary importance to solar cells. We find that for the SiNx prepared in this work, 1) S eff does not correlate universally with the bulk structural and optical properties such as chemical bond densities and refractive index, and 2) S eff depends primarily on the defect density at the SiNx-Si interface rather than the insulator charge. Finally, employing the optimized deposition condition, we achieve a relatively constant and low S eff,UL on low-resistivity (≤1.1 Ωcm) p- and n-type c-Si substrates over a broad range of n = 1.85–4.07. The results of this study demonstrate that the trade-off between optical transmission and surface passivation can be circumvented. Although we focus on photovoltaic applications, this study may be useful for any device for which it is desirable to maximize light transmission and surface passivation.This work was supported by an Australian Research Council Linkage between The Australian National University and Braggone Oy under Grant LP0989593

Satisfiability and Evolution

We show that, if truth assignments on $n$ variables reproduce through recombination so that satisfaction of a particular Boolean function confers a small evolutionary advantage, then a polynomially large population over polynomially many generations (polynomial in $n$ and the inverse of the initial satisfaction probability) will end up almost certainly consisting exclusively of satisfying truth assignments. We argue that this theorem sheds light on the problem of novelty in Evolution

On the Distribution of the Fourier Spectrum of Halfspaces

Bourgain showed that any noise stable Boolean function $f$ can be well-approximated by a junta. In this note we give an exponential sharpening of the parameters of Bourgain's result under the additional assumption that $f$ is a halfspace

Approximate resilience, monotonicity, and the complexity of agnostic learning

A function $f$ is $d$-resilient if all its Fourier coefficients of degree at most $d$ are zero, i.e., $f$ is uncorrelated with all low-degree parities. We study the notion of $\mathit{approximate}$ $\mathit{resilience}$ of Boolean functions, where we say that $f$ is $\alpha$-approximately $d$-resilient if $f$ is $\alpha$-close to a $[-1,1]$-valued $d$-resilient function in $\ell_1$ distance. We show that approximate resilience essentially characterizes the complexity of agnostic learning of a concept class $C$ over the uniform distribution. Roughly speaking, if all functions in a class $C$ are far from being $d$-resilient then $C$ can be learned agnostically in time $n^{O(d)}$ and conversely, if $C$ contains a function close to being $d$-resilient then agnostic learning of $C$ in the statistical query (SQ) framework of Kearns has complexity of at least $n^{\Omega(d)}$. This characterization is based on the duality between $\ell_1$ approximation by degree-$d$ polynomials and approximate $d$-resilience that we establish. In particular, it implies that $\ell_1$ approximation by low-degree polynomials, known to be sufficient for agnostic learning over product distributions, is in fact necessary. Focusing on monotone Boolean functions, we exhibit the existence of near-optimal $\alpha$-approximately $\widetilde{\Omega}(\alpha\sqrt{n})$-resilient monotone functions for all $\alpha>0$. Prior to our work, it was conceivable even that every monotone function is $\Omega(1)$-far from any $1$-resilient function. Furthermore, we construct simple, explicit monotone functions based on ${\sf Tribes}$ and ${\sf CycleRun}$ that are close to highly resilient functions. Our constructions are based on a fairly general resilience analysis and amplification. These structural results, together with the characterization, imply nearly optimal lower bounds for agnostic learning of monotone juntas