Unbalanced Expander Research Articles

Sampling Graphs without Forbidden Subgraphs and Unbalanced Expanders with Negligible Error

Sampling Graphs without Forbidden Subgraphs and Unbalanced Expanders with Negligible Error

Spatio-Temporal Compressive Sensing-Based Data Gathering in Wireless Sensor Networks

Sensory data in many wireless sensor networks feature spatio-temporal correlations, and compressive sensing (CS) plays an important role in energy-efficient data gathering. In this letter, we design a new CS-based data gathering algorithm, utilizing random sampling and random walks to select sensory data in temporal and spatial domains, respectively. Each measurement is obtained by summing the selected data. A novel sensing matrix is also designed based on the adjacency matrix of an unbalanced expander graph. Simulation shows that our proposed algorithm reduces energy consumption by up to 50.0% compared to the existing algorithms in a daily sea surface temperature measurement scenario.

Time for dithering: fast and quantized random embeddings via the restricted isometry property

Recently, many works have focused on the characterization of non-linear dimensionality reduction methods obtained by quantizing linear embeddings, e.g., to reach fast processing time, efficient data compression procedures, novel geometry-preserving embeddings or to estimate the information/bits stored in this reduced data representation. In this work, we prove that many linear maps known to respect the restricted isometry property (RIP) can induce a quantized random embedding with controllable multiplicative and additive distortions with respect to the pairwise distances of the data points beings considered. In other words, linear matrices having fast matrix-vector multiplication algorithms (e.g., based on partial Fourier ensembles or on the adjacency matrix of unbalanced expanders) can be readily used in the definition of fast quantized embeddings with small distortions. This implication is made possible by applying right after the linear map an additive and random dither that stabilizes the impact of the uniform scalar quantization operator applied afterwards. For different categories of RIP matrices, i.e., for different linear embeddings of a metric space $(\mathcal K \subset \mathbb R^n, \ell_q)$ in $(\mathbb R^m, \ell_p)$ with $p,q \geq 1$, we derive upper bounds on the additive distortion induced by quantization, showing that it decays either when the embedding dimension $m$ increases or when the distance of a pair of embedded vectors in $\mathcal K$ decreases. Finally, we develop a novel bi-dithered quantization scheme, which allows for a reduced distortion that decreases when the embedding dimension grows and independently of the considered pair of vectors.

For-All Sparse Recovery in Near-Optimal Time

An approximate sparse recovery system in ℓ 1 norm consists of parameters k , ϵ, N ; an m -by- N measurement Φ; and a recovery algorithm R . Given a vector, x , the system approximates x by xˆ = R (Φ x ), which must satisfy ‖ xˆ- x ‖ 1 ≤ (1+ϵ)‖ x - x k ‖ 1 . We consider the “for all” model, in which a single matrix Φ, possibly “constructed” non-explicitly using the probabilistic method, is used for all signals x . The best existing sublinear algorithm by Porat and Strauss [2012] uses O (ϵ −3 k log ( N / k )) measurements and runs in time O ( k 1 − α N α ) for any constant α > 0. In this article, we improve the number of measurements to O (ϵ − 2 k log ( N / k )), matching the best existing upper bound (attained by super-linear algorithms), and the runtime to O ( k 1+β poly(log N ,1/ϵ)), with a modest restriction that k ⩽ N 1 − α and ϵ ⩽ (log k /log N ) γ for any constants α, β, γ > 0. When k ⩽ log c N for some c > 0, the runtime is reduced to O ( k poly( N ,1/ϵ)). With no restrictions on ϵ, we have an approximation recovery system with m = O ( k /ϵlog ( N / k )((log N /log k ) γ + 1/ϵ)) measurements. The overall architecture of this algorithm is similar to that of Porat and Strauss [2012] in that we repeatedly use a weak recovery system (with varying parameters) to obtain a top-level recovery algorithm. The weak recovery system consists of a two-layer hashing procedure (or with two unbalanced expanders for a deterministic algorithm). The algorithmic innovation is a novel encoding procedure that is reminiscent of network coding and that reflects the structure of the hashing stages. The idea is to encode the signal position index i by associating it with a unique message m i , which will be encoded to a longer message m ′ i (in contrast to Porat and Strauss [2012] in which the encoding is simply the identity). Portions of the message m ′ i correspond to repetitions of the hashing, and we use a regular expander graph to encode the linkages among these portions. The decoding or recovery algorithm consists of recovering the portions of the longer messages m ′ i and then decoding to the original messages m i , all the while ensuring that corruptions can be detected and/or corrected. The recovery algorithm is similar to list recovery introduced in Indyk et al. [2010] and used in Gilbert et al. [2013]. In our algorithm, the messages { m i } are independent of the hashing, which enables us to obtain a better result.

Unbalanced Expander Based Compressive Data Gathering in Clustered Wireless Sensor Networks

Conventional compressive sensing-based data gathering (CS-DG) algorithms require a large number of sensors for each compressive sensing measurement, thereby resulting in high energy consumption in clustered wireless sensor networks (WSNs). To solve this problem, we propose a novel energy-efficient CS-DG algorithm, which exploits the better reconstruction accuracy of the adjacency matrix of an unbalanced expander graph. In the proposed CS-DG algorithm, each measurement is the sum of a few sensory data, which are jointly determined by random sampling and random walks. Through theoretical analysis, we prove that the constructed M × N sparse binary sensing matrix is the adjacency matrix of a (k, ε) unbalanced expander graph when M = O (k log N/k) and t = O (N <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">c</sub> /(kq)) for WSNs with Nc clusters, where 0 ≤ q ≤ 1 and N <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">c</sub> > k. Simulation results show our proposed CS-DG has better performance than existing algorithms in terms of reconstruction accuracy and energy consumption. When hybrid energy-efficient distributed clustering algorithm is used, to achieve the same reconstruction accuracy, our proposed CS-DG can save energy by at least 27.8%.

On Rigid Matrices and U-Polynomials

We introduce a class of polynomials, which we call U-polynomials, and show that the problem of explicitly constructing a rigid matrix can be reduced to the problem of explicitly constructing a small hitting set for this class. We prove that small-bias sets are hitting sets for the class of U-polynomials, though their size is larger than desired. Furthermore, we give two alternative proofs for the fact that small-bias sets induce rigid matrices. Finally, we construct rigid matrices from unbalanced expanders, with essentially the same size as the construction via small-bias sets.

Optimal Designs for Lasso and Dantzig Selector Using Expander Codes

We investigate the high-dimensional regression problem using adjacency matrices of unbalanced expander graphs. In this frame, we prove that the ℓ <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> -prediction error and ℓ <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> -risk of the lasso, and the Dantzig selector are optimal up to an explicit multiplicative constant. Thus, we can estimate a high-dimensional target vector with an error term similar to the one obtained in a situation where one knows the support of the largest coordinates in advance. Moreover, we show that these design matrices have an explicit restricted eigenvalue. Precisely, they satisfy the restricted eigenvalue assumption and compatibility condition with an explicit constant. Eventually, we capitalize on the recent construction of unbalanced expander graphs due to Guruswami, Umans, and Vadhan, to provide a deterministic polynomial time construction of these design matrices.

Kakeya Sets, New Mergers, and Old Extractors

A merger is a probabilistic procedure which extracts the randomness out of any (arbitrarily correlated) set of random variables, as long as one of them is uniform. Our main result is an efficient, simple, and optimal (to constant factors) merger, which, for $k$ random variables on $n$ bits each, uses an $O(\log(nk))$ seed, and whose error is $1/nk$. Our merger can be viewed as a derandomized version of the merger of Lu et al. [Extractors: Optimal up to constant factors, in Proceedings of the 35th Annual ACM Symposium on Theory of Computing, ACM, New York, 2003, pp. 602-611]. Its analysis generalizes the recent resolution of the Kakeya problem in finite fields of Dvir [J. Amer. Math. Soc., 22 (2009), pp. 1093-1097]. Following the plan set forth by Ta-Shma [Refining Randomness, Ph.D. thesis, The Hebrew University, Jerusalem, Israel, 1996] who defined mergers as part of this plan, our merger provides the last “missing link” to a simple and modular construction of extractors for all entropies, which is optimal to constant factors in all parameters. This complements the elegant construction of such extractors given by Guruswami, Umans, and Vadhan [Unbalanced expanders and randomness extractors from Parvaresh-Vardy codes, in CCC '07: Proceedings of the Twenty-Second Annual IEEE Conference on Computaional Complexity, IEEE Computer Society, Washington, DC, 2007, pp. 96-108]. We also give simple extensions of our merger in two directions. First, we generalize it to handle the case where no source is uniform—in that case the merger will extract the entropy present in the most random of the given sources. Second, we observe that the merger works just as well in the computational setting, when the sources are efficiently samplable, and computational notions of entropy replace the information theoretic ones.

Unbalanced expanders and randomness extractors from Parvaresh--Vardy codes

We give an improved explicit construction of highly unbalanced bipartite expander graphs with expansion arbitrarily close to the degree (which is polylogarithmic in the number of vertices). Both the degree and the number of right-hand vertices are polynomially close to optimal, whereas the previous constructions of Ta-Shma et al. [2007] required at least one of these to be quasipolynomial in the optimal. Our expanders have a short and self-contained description and analysis, based on the ideas underlying the recent list-decodable error-correcting codes of Parvaresh and Vardy [2005]. Our expanders can be interpreted as near-optimal “randomness condensers,” that reduce the task of extracting randomness from sources of arbitrary min-entropy rate to extracting randomness from sources of min-entropy rate arbitrarily close to 1, which is a much easier task. Using this connection, we obtain a new, self-contained construction of randomness extractors that is optimal up to constant factors, while being much simpler than the previous construction of Lu et al. [2003] and improving upon it when the error parameter is small (e.g., 1/poly(n)).

The complexity of the matroid–greedoid partition problem

We show that the maximum matroid–greedoid partition problem is NP-hard to approximate to within 1 / 2 + ε for any ε > 0 , which matches the trivial factor 1/2 approximation algorithm. The main tool in our hardness of approximation result is an extractor code with polynomial rate, alphabet size and list size, together with an efficient algorithm for list-decoding. We show that the recent extractor construction of Guruswami, Umans and Vadhan [V. Guruswami, C. Umans, S.P. Vadhan, Unbalanced expanders and randomness extractors from Parvaresh-Vardy codes, in: IEEE Conference on Computational Complexity, IEEE Computer Society, 2007, pp. 96–108] can be used to obtain a code with these properties. We also show that the parameterized matroid–greedoid partition problem is fixed-parameter tractable.

Storing information with extractors

We deal with the problem of storing a set of K elements that are taken from a large universe of size N, such that membership in the set can be determined with high probability by looking at just one bit of the representation. Buhrman et al. show an explicit construction with about K 2log N storing bits. We show an explicit construction with about K 1+o(1) storing bits, that gets closer to the optimal Klog N bound. Our technique is of independent interest. Buhrman et al. show a non-explicit optimal (up to constant factors) construction that is based on the existence of certain good unbalanced expanders. To make the construction explicit one needs to be able to explicitly ‘encode’ and ‘decode’ such expanding graphs. We generalize the notion of loss-less condensers of Ta-Shma et al. [Proc. 33rd Annual ACM Symposium on Theory of Computing, 2001, pp. 143–152] and build such graphs. We further show how to efficiently decode such graphs using an observation from Ta-Shma and Zuckerman [Manuscript, 2001] about Trevisan's extractor.