MAC with Action-Dependent State Information at One Encoder

Problems dealing with the ability to take an action that affects the states of state-dependent communication channels are of timely interest and importance. Therefore, we extend the study of action-dependent channels, which until now focused on point-to-point models, to multiple-access channels (MAC). In this paper, we consider a two-user, state-dependent MAC, in which one of the encoders, called the informed encoder, is allowed to take an action that affects the formation of the channel states. Two independent messages are to be sent through the channel: a common message known to both encoders and a private message known only to the informed encoder. In addition, the informed encoder has access to the sequence of channel states in a non-causal manner. Our framework generalizes previously evaluated settings of state dependent point-to-point channels with actions and MACs with common messages. We derive a single letter characterization of the capacity region for this setting. Using this general result, we obtain and compute the capacity region for the Gaussian action-dependent MAC. The unique methods used in solving the Gaussian case are then applied to obtain the capacity of the Gaussian action-dependent point-to-point channel; a problem was left open until this work. Finally, we establish some dualities between action-dependent channel coding and source coding problems. Specifically, we obtain a duality between the considered MAC setting and the rate distortion model known as "Successive Refinement with Actions". This is done by developing a set of simple duality principles that enable us to successfully evaluate the outcome of one problem given the other.Comment: 1. Parts of this paper appeared in the IEEE International Symposium on Information Theory (ISIT 2012),Cambridge, MA, US, July 2012 and at the IEEE 27th Convention of Electrical and Electronics Engineers in Israel (IEEEI 2012), Nov. 2012. 2. This work has been supported by the CORNET Consortium Israel Ministry for Industry and Commerc

New High Dimensional Expanders from Covers

We present a new construction of high dimensional expanders based on covering spaces of simplicial complexes. High dimensional expanders (HDXs) are hypergraph analogues of expander graphs. They have many uses in theoretical computer science, but unfortunately only few constructions are known which have arbitrarily small local spectral expansion. We give a randomized algorithm that takes as input a high dimensional expander $X$ (satisfying some mild assumptions). It outputs a sub-complex $Y \subseteq X$ that is a high dimensional expander and has infinitely many simplicial covers. These covers form new families of bounded-degree high dimensional expanders. The sub-complex $Y$ inherits $X$'s underlying graph and its links are sparsifications of the links of $X$. When the size of the links of $X$ is $O(\log |X|)$, this algorithm can be made deterministic. Our algorithm is based on the groups and generating sets discovered by Lubotzky, Samuels and Vishne (2005), that were used to construct the first discovered high dimensional expanders. We show these groups give rise to many more ``randomized'' high dimensional expanders. In addition, our techniques also give a random sparsification algorithm for high dimensional expanders, that maintains its local spectral properties. This may be of independent interest

Coboundary and cosystolic expansion without dependence on dimension or degree

We give new bounds on the cosystolic expansion constants of several families of high dimensional expanders, and the known coboundary expansion constants of order complexes of homogeneous geometric lattices, including the spherical building of $SL_n(F_q)$. The improvement applies to the high dimensional expanders constructed by Lubotzky, Samuels and Vishne, and by Kaufman and Oppenheim. Our new expansion constants do not depend on the degree of the complex nor on its dimension, nor on the group of coefficients. This implies improved bounds on Gromov's topological overlap constant, and on Dinur and Meshulam's cover stability, which may have applications for agreement testing. In comparison, existing bounds decay exponentially with the ambient dimension (for spherical buildings) and in addition decay linearly with the degree (for all known bounded-degree high dimensional expanders). Our results are based on several new techniques: * We develop a new "color-restriction" technique which enables proving dimension-free expansion by restricting a multi-partite complex to small random subsets of its color classes. * We give a new "spectral" proof for Evra and Kaufman's local-to-global theorem, deriving better bounds and getting rid of the dependence on the degree. This theorem bounds the cosystolic expansion of a complex using coboundary expansion and spectral expansion of the links. * We derive absolute bounds on the coboundary expansion of the spherical building (and any order complex of a homogeneous geometric lattice) by constructing a novel family of very short cones

Chernoff Bounds and Reverse Hypercontractivity on HDX

We prove optimal concentration of measure for lifted functions on high dimensional expanders (HDX). Let $X$ be a $k$-dimensional HDX. We show for any $i\leq k$ and $f:X(i)\to [0,1]$: $$\Pr_{s\in X(k)}\left[\left|\underset{{t\subseteq s}}{\mathbb{E}}[f(t)]-\mu\right|\geq\varepsilon\right]\leq exp\left(-\varepsilon^2\frac{k}{i}\right).$$ Using this fact, we prove that high dimensional expanders are reverse hypercontractive, a powerful functional inequality from discrete analysis implying that for any sets $A,B \subset X(k)$, the probability a $\rho$-correlated pair passes between them is at least $$\Pr_{s,s' \sim T_\rho}[s \in A, s' \in B] \geq \Pr[A]^{O(1)} \Pr[B]^{O(1)}.$$ Our results hold under weak spectral assumptions on $X$. Namely we prove exponential concentration of measure for any complex below the `Trickling-Down Threshold' (beyond which concentration may be arbitrarily poor), and optimal concentration for $\sqrt{k}$-skeletons of such complexes. We also show optimal bounds for the top dimension of stronger HDX among other settings. We leverage our inequalities to prove several new agreement testing theorems on high dimensional expanders, including a new 99%-regime test for subsets, and a variant of the `Z-test' achieving inverse exponential soundness under the stronger assumption of $\ell_\infty$-expansion. The latter gives rise to the first optimal testers beyond the complete complex and products, a stepping stone toward the use of HDX in strong soundness PCPs. We also give applications within expansion, analysis, combinatorics, and coding theory, including a proof that two-sided HDX have optimal geometric overlap (giving the first explicit bounded-degree construction), near-optimal double samplers, new super-exponential degree lower bounds for certain HDX, distance-amplified list-decodable and locally testable codes, a Frankl-R\"odl Theorem and more.Comment: A mistake in the proof of Theorem 7.14 was corrected (Theorem 7.15 in previous version). Also some minor changes and typos were fixe

Coboundary and Cosystolic Expansion Without Dependence on Dimension or Degree

We give new bounds on the cosystolic expansion constants of several families of high dimensional expanders, and the known coboundary expansion constants of order complexes of homogeneous geometric lattices, including the spherical building of SL_n(_q). The improvement applies to the high dimensional expanders constructed by Lubotzky, Samuels and Vishne, and by Kaufman and Oppenheim. Our new expansion constants do not depend on the degree of the complex nor on its dimension, nor on the group of coefficients. This implies improved bounds on Gromov’s topological overlap constant, and on Dinur and Meshulam’s cover stability, which may have applications for agreement testing. In comparison, existing bounds decay exponentially with the ambient dimension (for spherical buildings) and in addition decay linearly with the degree (for all known bounded-degree high dimensional expanders). Our results are based on several new techniques: - We develop a new "color-restriction" technique which enables proving dimension-free expansion by restricting a multi-partite complex to small random subsets of its color classes. - We give a new "spectral" proof for Evra and Kaufman’s local-to-global theorem, deriving better bounds and getting rid of the dependence on the degree. This theorem bounds the cosystolic expansion of a complex using coboundary expansion and spectral expansion of the links. - We derive absolute bounds on the coboundary expansion of the spherical building (and any order complex of a homogeneous geometric lattice) by constructing a novel family of very short cones

Swap cosystolic expansion

We introduce and study swap cosystolic expansion, a new expansion property of simplicial complexes. We prove lower bounds for swap coboundary expansion of spherical buildings and use them to lower bound swap cosystolic expansion of the LSV Ramanujan complexes. Our motivation is the recent work (in a companion paper) showing that swap cosystolic expansion implies agreement theorems. Together the two works show that these complexes support agreement tests in the low acceptance regime. Swap cosystolic expansion is defined by considering, for a given complex $X$, its faces complex $F^r X$, whose vertices are $r$-faces of $X$ and where two vertices are connected if their disjoint union is also a face in $X$. The faces complex $F^r X$ is a derandomizetion of the product of $X$ with itself $r$ times. The graph underlying $F^rX$ is the swap walk of $X$, known to have excellent spectral expansion. The swap cosystolic expansion of $X$ is defined to be the cosystolic expansion of $F^r X$. Our main result is a $\exp(-O(\sqrt r))$ lower bound on the swap coboundary expansion of the spherical building and the swap cosystolic expansion of the LSV complexes. For more general coboundary expanders we show a weaker lower bound of $exp(-O(r))$

Impact of Age and Body Site on Adult Female Skin Surface pH

Background: pH is known as an important parameter in epidermal barrier function and homeostasis. Aim: The impact of age and body site on skin surface pH (pH(SS)) of women was evaluated in vivo. Methods: Time domain dual lifetime referencing with luminescent sensor foils was used for pH(SS) measurements. pH(SS) was measured on the forehead, the temple, and the volar forearm of adult females (n = 97, 52.87 +/- 18.58 years, 20-97 years). Every single measurement contained 2,500 pH values due to the luminescence imaging technique used. Results: pH(SS) slightly increases with age on all three investigated body sites. There are no significant differences in pH(SS) between the three investigated body sites. Conclusion: Adult pH(SS) on the forehead, the temple and the volar forearm increases slightly with age. This knowledge is crucial for adapting medical skin care products. Copyright (C) 2012 S. Karger AG, Base

Sparse High Dimensional Expanders via Local Lifts

High dimensional expanders (HDXs) are a hypergraph generalization of expander graphs. They are extensively studied in the math and TCS communities due to their many applications. Like expander graphs, HDXs are especially interesting for applications when they are bounded degree, namely, if the number of edges adjacent to every vertex is bounded. However, only a handful of constructions are known to have this property, all of which rely on algebraic techniques. In particular, no random or combinatorial construction of bounded degree high dimensional expanders is known. As a result, our understanding of these objects is limited. The degree of an i-face in an HDX is the number of (i+1)-faces that contain it. In this work we construct complexes whose higher dimensional faces have bounded degree. This is done by giving an elementary and deterministic algorithm that takes as input a regular k-dimensional HDX X and outputs another regular k-dimensional HDX X̂ with twice as many vertices. While the degree of vertices in X̂ grows, the degree of the (k-1)-faces in X̂ stays the same. As a result, we obtain a new "algebra-free" construction of HDXs whose (k-1)-face degree is bounded. Our construction algorithm is based on a simple and natural generalization of the expander graph construction by Bilu and Linial [Yehonatan Bilu and Nathan Linial, 2006], which build expander graphs using lifts coming from edge signings. Our construction is based on local lifts of high dimensional expanders, where a local lift is a new complex whose top-level links are lifts of the links of the original complex. We demonstrate that a local lift of an HDX is also an HDX in many cases. In addition, combining local lifts with existing bounded degree constructions creates new families of bounded degree HDXs with significantly different links than before. For every large enough D, we use this technique to construct families of bounded degree HDXs with links that have diameter ≥ D

Sparse High Dimensional Expanders via Local Lifts

High dimensional expanders (HDXs) are a hypergraph generalization of expander graphs. They are extensively studied in the math and TCS communities due to their many applications. Like expander graphs, HDXs are especially interesting for applications when they are bounded degree, namely, if the number of edges adjacent to every vertex is bounded. However, only a handful of constructions are known to have this property, all of which rely on algebraic techniques. In particular, no random or combinatorial construction of bounded degree HDXs is known. As a result, our understanding of these objects is limited. The degree of an $i$-face in an HDX is the number of $(i+1)$-faces containing it. In this work we construct HDXs whose higher dimensional faces have bounded degree. This is done by giving an elementary and deterministic algorithm that takes as input a regular $k$-dimensional HDX $X$ and outputs another $k$-dimensional HDX $\widehat{X}$ with twice as many vertices. While the degree of vertices in $\widehat{X}$ grows, the degree of the $(k-1)$-faces in $\widehat{X}$ stays the same. As a result, we obtain a new `algebra-free' construction of HDXs whose $(k-1)$-face degree is bounded. Our algorithm is based on a simple and natural generalization of the construction by Bilu and Linial (Combinatorica, 2006), which build expanders using lifts coming from edge signings. Our construction is based on local lifts of HDXs, where a local lift is a complex whose top-level links are lifts of links in the original complex. We demonstrate that a local lift of an HDX is an HDX in many cases. In addition, combining local lifts with existing bounded degree constructions creates new families of bounded degree HDXs with significantly different links than before. For every large enough $D$, we use this technique to construct families of bounded degree HDXs with links that have diameter $\geq D$