OpenRaiser/Intern-Atlas huggingface dataset:
The Cell Probe Complexity of Dynamic Range Counting
Venue: arXiv cs.DS 2012
Year: 2012

In this paper we develop a new technique for proving lower bounds on the update time and query time of dynamic data structures in the cell probe model. With this technique, we prove the highest lower bound to date for any explicit problem, namely a lower bound of $t_q=Ω((\lg n/\lg(wt_u))^2)$. Here $n$ is the number of update operations, $w$ the cell size, $t_q$ the query time and $t_u$ the update time. In the most natural setting of cell size $w=Θ(\lg n)$, this gives a lower bound of $t_q=Ω((\lg n/\lg \lg n)^2)$ for any polylogarithmic update time. This bound is almost a quadratic improvement over the highest previous lower bound of $Ω(\lg n)$, due to Pǎtraşcu and Demaine [SICOMP'06]. We prove the lower bound for the fundamental problem of weighted orthogonal range counting. In this problem, we are to support insertions of two-dimensional points, each assigned a $Θ(\lg n)$-bit integer weight. A query to this problem is specified by a point $q=(x,y)$, and the goal is to report the sum of the weights assigned to the points dominated by $q$, where a point $(x',y')$ is dominated by $q$ if $x' \leq x$ and $y' \leq y$. In addition to being the highest cell probe lower bound to date, the lower bound is also tight for data structures with update time $t_u = Ω(\lg^{2+\eps}n)$, where $\eps>0$ is an arbitrarily small constant.
Achievable Rates of MIMO Systems with Linear Precoding and Iterative LMMSE Detection
Venue: arXiv cs.IT 2012
Year: 2012

We establish area theorems for iterative detection over coded linear systems (including multiple-input multipleoutput (MIMO) channels, inter-symbol-interference (ISI) channels, and orthogonal frequency-division multiplexing (OFDM) systems). We propose a linear precoding technique that asymptotically ensures the Gaussianness of the messages passed in iterative detection, as the transmission block length tends to infinity. We show that the proposed linear precoding scheme with iterative linear minimum mean-square error (LMMSE) detection is potentially information lossless, under various assumptions on the availability of channel state information at the transmitter (CSIT). Numerical results are provided to verify our analysis.
Hypercontractive Inequality for Pseudo-Boolean Functions of Bounded Fourier Width
Venue: arXiv cs.DM 2012
Year: 2012

A function $f:\ \{-1,1\}^n\rightarrow \mathbb{R}$ is called pseudo-Boolean. It is well-known that each pseudo-Boolean function $f$ can be written as $f(x)=\sum_{I\in {\cal F}}\hat{f}(I)χ_I(x),$ where ${\cal F}\subseteq \{I:\ I\subseteq [n]\}$, $[n]=\{1,2,...,n\}$, and $χ_I(x)=\prod_{i\in I}x_i$ and $\hat{f}(I)$ are non-zero reals. The degree of $f$ is $\max \{|I|:\ I\in {\cal F}\}$ and the width of $f$ is the minimum integer $ρ$ such that every $i\in [n]$ appears in at most $ρ$ sets in $\cal F$. For $i\in [n]$, let $\mathbf{x}_i$ be a random variable taking values 1 or -1 uniformly and independently from all other variables $\mathbf{x}_j$, $j\neq i.$ Let $\mathbf{x}=(\mathbf{x}_1,...,\mathbf{x}_n)$. The $p$-norm of $f$ is $||f||_p=(\mathbb E[|f(\mathbf{x})|^p])^{1/p}$ for any $p\ge 1$. It is well-known that $||f||_q\ge ||f||_p$ whenever $q> p\ge 1$. However, the higher norm can be bounded by the lower norm times a coefficient not directly depending on $f$: if $f$ is of degree $d$ and $q> p>1$ then $ ||f||_q\le (\frac{q-1}{p-1})^{d/2}||f||_p.$ This inequality is called the Hypercontractive Inequality. We show that one can replace $d$ by $ρ$ in the Hypercontractive Inequality for each $q> p\ge 2$ as follows: $ ||f||_q\le ((2r)!ρ^{r-1})^{1/(2r)}||f||_p,$ where $r=\lceil q/2\rceil$. For the case $q=4$ and $p=2$, which is important in many applications, we prove a stronger inequality: $ ||f||_4\le (2ρ+1)^{1/4}||f||_2.$
Security Notions for Information Theoretically Secure Encryptions
Venue: arXiv cs.CR 2012
Year: 2012

This paper is concerned with several security notions for information theoretically secure encryptions defined by the variational (statistical) distance. To ensure the perfect secrecy (PS), the mutual information is often used to evaluate the statistical independence between a message and a cryptogram. On the other hand, in order to recognize the information theoretically secure encryptions and computationally secure ones comprehensively, it is necessary to reconsider the notion of PS in terms of the variational distance. However, based on the variational distance, three kinds of definitions for PS are naturally introduced, but their relations are not known. In this paper, we clarify that one of three definitions for PS with the variational distance, which is a straightforward extension of Shannon's perfect secrecy, is stronger than the others, and the weaker two definitions of PS are essentially equivalent to the statistical versions of indistinguishability and semantic security.
Minimal resolving sets for the hypercube
Venue: arXiv cs.DM 2012
Year: 2012

For a given undirected graph $G$, an \emph{ordered} subset $S = {s_1,s_2,...,s_k} \subseteq V$ of vertices is a resolving set for the graph if the vertices of the graph are distinguishable by their vector of distances to the vertices in $S$. While a superset of any resolving set is always a resolving set, a proper subset of a resolving set is not necessarily a resolving set, and we are interested in determining resolving sets that are minimal or that are minimum (of minimal cardinality). Let $Q^n$ denote the $n$-dimensional hypercube with vertex set ${0,1}^n$. In Erdös and Renyi (Erdos & Renyi, 1963) it was shown that a particular set of $n$ vertices forms a resolving set for the hypercube. The main purpose of this note is to prove that a proper subset of that set of size $n-1$ is also a resolving set for the hypercube for all $n \ge 5$ and that this proper subset is a minimal resolving set.
Extended formulations for polygons
Venue: arXiv cs.DM 2012
Year: 2012

The extension complexity of a polytope $P$ is the smallest integer $k$ such that $P$ is the projection of a polytope $Q$ with $k$ facets. We study the extension complexity of $n$-gons in the plane. First, we give a new proof that the extension complexity of regular $n$-gons is $O(\log n)$, a result originating from work by Ben-Tal and Nemirovski (2001). Our proof easily generalizes to other permutahedra and simplifies proofs of recent results by Goemans (2009), and Kaibel and Pashkovich (2011). Second, we prove a lower bound of $\sqrt{2n}$ on the extension complexity of generic $n$-gons. Finally, we prove that there exist $n$-gons whose vertices lie on a $O(n) \times O(n^2)$ integer grid with extension complexity $Ω(\sqrt{n}/\sqrt{\log n})$.
Vision-Based Navigation I: A navigation filter for fusing DTM/correspondence updates
Venue: arXiv cs.CV 2012
Year: 2012

An algorithm for pose and motion estimation using corresponding features in images and a digital terrain map is proposed. Using a Digital Terrain (or Digital Elevation) Map (DTM/DEM) as a global reference enables recovering the absolute position and orientation of the camera. In order to do this, the DTM is used to formulate a constraint between corresponding features in two consecutive frames. The utilization of data is shown to improve the robustness and accuracy of the inertial navigation algorithm. Extended Kalman filter was used to combine results of inertial navigation algorithm and proposed vision-based navigation algorithm. The feasibility of this algorithms is established through numerical simulations.
On the Furthest Hyperplane Problem and Maximal Margin Clustering
Venue: arXiv cs.CC 2012
Year: 2012

This paper introduces the Furthest Hyperplane Problem (FHP), which is an unsupervised counterpart of Support Vector Machines. Given a set of n points in Rd, the objective is to produce the hyperplane (passing through the origin) which maximizes the separation margin, that is, the minimal distance between the hyperplane and any input point. To the best of our knowledge, this is the first paper achieving provable results regarding FHP. We provide both lower and upper bounds to this NP-hard problem. First, we give a simple randomized algorithm whose running time is n^O(1/θ^2) where θ is the optimal separation margin. We show that its exponential dependency on 1/θ^2 is tight, up to sub-polynomial factors, assuming SAT cannot be solved in sub-exponential time. Next, we give an efficient approxima- tion algorithm. For any α \in [0, 1], the algorithm produces a hyperplane whose distance from at least 1 - 5α fraction of the points is at least α times the optimal separation margin. Finally, we show that FHP does not admit a PTAS by presenting a gap preserving reduction from a particular version of the PCP theorem.
Fluctuations of spiked random matrix models and failure diagnosis in sensor networks
Venue: arXiv cs.IT 2012
Year: 2012

In this article, the joint fluctuations of the extreme eigenvalues and eigenvectors of a large dimensional sample covariance matrix are analyzed when the associated population covariance matrix is a finite-rank perturbation of the identity matrix, corresponding to the so-called spiked model in random matrix theory. The asymptotic fluctuations, as the matrix size grows large, are shown to be intimately linked with matrices from the Gaussian unitary ensemble (GUE). When the spiked population eigenvalues have unit multiplicity, the fluctuations follow a central limit theorem. This result is used to develop an original framework for the detection and diagnosis of local failures in large sensor networks, for known or unknown failure magnitude.
The Divergence of Reinforcement Learning Algorithms with Value-Iteration and Function Approximation
Venue: arXiv cs.LG 2012
Year: 2012

This paper gives specific divergence examples of value-iteration for several major Reinforcement Learning and Adaptive Dynamic Programming algorithms, when using a function approximator for the value function. These divergence examples differ from previous divergence examples in the literature, in that they are applicable for a greedy policy, i.e. in a "value iteration" scenario. Perhaps surprisingly, with a greedy policy, it is also possible to get divergence for the algorithms TD(1) and Sarsa(1). In addition to these divergences, we also achieve divergence for the Adaptive Dynamic Programming algorithms HDP, DHP and GDHP.
On the Achievability of Interference Alignment for Three-Cell Constant Cellular Interfering Networks
Venue: arXiv cs.IT 2012
Year: 2012

For a three-cell constant cellular interfering network, a new property of alignment is identified, i.e., interference alignment (IA) solution obtained in an user-cooperation scenario can also be applied in a non-cooperation environment. By using this property, an algorithm is proposed by jointly designing transmit and receive beamforming matrices. Analysis and numerical results show that more degree of freedom (DoF) can be achieved compared with conventional schemes in most cases.
Multivalued Subsets Under Information Theory
Venue: arXiv cs.IT 2012
Year: 2012

In the fields of finance, engineering and sciences data mining/ machine learning has held an eminent position in predictive analysis. Complex algorithms and adaptive decision models have contributed towards streamlining research as well as improve forecasting. Extensive study in areas surrounding computation and mathematical sciences has primarily been responsible for the field's development. Classification based modeling, which holds a prominent position amongst the different rule-based algorithms, is one of the most widely used decision making tool. The decision tree has a place of profound significance in classification modeling. A number of heuristics have been developed over the years to refine its decision making process. Most heuristics applied to such tree-based learning algorithms derive their roots from Shannon's 'Information Theory'. The current application of this theory is directed towards individual assessment of the attribute-values. The proposed study takes a look at the effects of combining these values with the aim to improve the 'Information Gain'. A search-based heuristic tool is applied for identifying the subsets sharing a better gain value than the ones presented in the GID3 approach. An application towards the feature selection stage of the mining process has been tested and presented with statistical analysis.
The Channel Capacity Increases with Power
Venue: arXiv cs.IT 2012
Year: 2012

It is proved that for memoryless vector channels, maximizing the mutual information over all source distributions with a certain average power or over the larger set of source distributions with upperbounded average power yields the same channel capacity in both cases. Hence, the channel capacity cannot decrease with increasing average transmitted power, not even for channels with severe nonlinear distortion.
Towards an Optimal Space-and-Query-Time Index for Top-k Document Retrieval
Venue: arXiv cs.DS 2012
Year: 2012

Let $\D = $$ \{d_1,d_2,...d_D\}$ be a given set of $D$ string documents of total length $n$, our task is to index $\D$, such that the $k$ most relevant documents for an online query pattern $P$ of length $p$ can be retrieved efficiently. We propose an index of size $|CSA|+n\log D(2+o(1))$ bits and $O(t_{s}(p)+k\log\log n+poly\log\log n)$ query time for the basic relevance metric \emph{term-frequency}, where $|CSA|$ is the size (in bits) of a compressed full text index of $\D$, with $O(t_s(p))$ time for searching a pattern of length $p$ . We further reduce the space to $|CSA|+n\log D(1+o(1))$ bits, however the query time will be $O(t_s(p)+k(\log σ\log\log n)^{1+ε}+poly\log\log n)$, where $σ$ is the alphabet size and $ε>0$ is any constant.
Robust Coding for Lossy Computing with Receiver-Side Observation Costs
Venue: arXiv cs.IT 2012
Year: 2012

An encoder wishes to minimize the bit rate necessary to guarantee that a decoder is able to calculate a symbolwise function of a sequence available only at the encoder and a sequence that can be measured only at the decoder. This classical problem, first studied by Yamamoto, is addressed here by including two new aspects: (i) The decoder obtains noisy measurements of its sequence, where the quality of such measurements can be controlled via a cost-constrained "action" sequence; (ii) Measurement at the decoder may fail in a way that is unpredictable to the encoder, thus requiring robust encoding. The considered scenario generalizes known settings such as the Heegard-Berger-Kaspi and the "source coding with a vending machine" problems. The rate-distortion-cost function is derived and numerical examples are also worked out to obtain further insight into the optimal system design.
Entropy Density and Mismatch in High-Rate Scalar Quantization with Renyi Entropy Constraint
Venue: arXiv cs.IT 2012
Year: 2012

Properties of scalar quantization with $r$th power distortion and constrained Rényi entropy of order $α\in (0,1)$ are investigated. For an asymptotically (high-rate) optimal sequence of quantizers, the contribution to the Rényi entropy due to source values in a fixed interval is identified in terms of the "entropy density" of the quantizer sequence. This extends results related to the well-known point density concept in optimal fixed-rate quantization. A dual of the entropy density result quantifies the distortion contribution of a given interval to the overall distortion. The distortion loss resulting from a mismatch of source densities in the design of an asymptotically optimal sequence of quantizers is also determined. This extends Bucklew's fixed-rate ($α=0$) and Gray \emph{et al.}'s variable-rate ($α=1$) mismatch results to general values of the entropy order parameter $α$.
Turbo Lattices: Construction and Error Decoding Performance
Venue: arXiv cs.IT 2012
Year: 2012

In this paper a new class of lattices called turbo lattices is introduced and established. We use the lattice Construction D to produce turbo lattices. This method needs a set of nested linear codes as its underlying structure. We benefit from turbo codes as our basis codes. Therefore, a set of nested turbo codes based on nested interleavers (block interleavers) and nested convolutional codes is built. To this end, we employ both tail-biting and zero-tail convolutional codes. Using these codes, along with construction D, turbo lattices are created. Several properties of Construction D lattices and fundamental characteristics of turbo lattices including the minimum distance, coding gain and kissing number are investigated. Furthermore, a multi-stage turbo lattice decoding algorithm based on iterative turbo decoding algorithm is given. We show, by simulation, that turbo lattices attain good error performance within $\sim1.25 dB$ from capacity at block length of $n=1035$. Also an excellent performance of only $\sim.5 dB$ away from capacity at SER of $10^{-5}$ is achieved for size $n=10131$.
Equilibria und weiteres Heiteres II
Venue: arXiv cs.LO 2012
Year: 2012

We investigate several technical and conceptual questions. Our main subject is the investigation of independence as a ternary relation in the context of non-monotonic logic. In the context of probability, this investigation was started by W.Spohn et al., and then followed by J.Pearl. We look at products of function sets, and thus continue our own investigation of independence in non-monotonic logic. We show that a finite characterization of this relation in our context is impossible, and indicate how to construct all valid rules.
The expressibility of functions on the Boolean domain, with applications to Counting CSPs
Venue: arXiv cs.CC 2012
Year: 2012

An important tool in the study of the complexity of Constraint Satisfaction Problems (CSPs) is the notion of a relational clone, which is the set of all relations expressible using primitive positive formulas over a particular set of base relations. Post's lattice gives a complete classification of all Boolean relational clones, and this has been used to classify the computational difficulty of CSPs. Motivated by a desire to understand the computational complexity of (weighted) counting CSPs, we develop an analogous notion of functional clones and study the landscape of these clones. One of these clones is the collection of log-supermodular (lsm) functions, which turns out to play a significant role in classifying counting CSPs. In the conservative case (where all nonnegative unary functions are available), we show that there are no functional clones lying strictly between the clone of lsm functions and the total clone (containing all functions). Thus, any counting CSP that contains a single nontrivial non-lsm function is computationally as hard to approximate as any problem in #P. Furthermore, we show that any non-trivial functional clone (in a sense that will be made precise) contains the binary function "implies". As a consequence, in the conservative case, all non-trivial counting CSPs are as hard as #BIS, the problem of counting independent sets in a bipartite graph. Given the complexity-theoretic results, it is natural to ask whether the "implies" clone is equivalent to the clone of lsm functions. We use the Mobius transform and the Fourier transform to show that these clones coincide precisely up to arity 3. It is an intriguing open question whether the lsm clone is finitely generated. Finally, we investigate functional clones in which only restricted classes of unary functions are available.
Local Component Analysis
Venue: arXiv cs.LG 2012
Year: 2012

Kernel density estimation, a.k.a. Parzen windows, is a popular density estimation method, which can be used for outlier detection or clustering. With multivariate data, its performance is heavily reliant on the metric used within the kernel. Most earlier work has focused on learning only the bandwidth of the kernel (i.e., a scalar multiplicative factor). In this paper, we propose to learn a full Euclidean metric through an expectation-minimization (EM) procedure, which can be seen as an unsupervised counterpart to neighbourhood component analysis (NCA). In order to avoid overfitting with a fully nonparametric density estimator in high dimensions, we also consider a semi-parametric Gaussian-Parzen density model, where some of the variables are modelled through a jointly Gaussian density, while others are modelled through Parzen windows. For these two models, EM leads to simple closed-form updates based on matrix inversions and eigenvalue decompositions. We show empirically that our method leads to density estimators with higher test-likelihoods than natural competing methods, and that the metrics may be used within most unsupervised learning techniques that rely on such metrics, such as spectral clustering or manifold learning methods. Finally, we present a stochastic approximation scheme which allows for the use of this method in a large-scale setting.
Throughput Maximization for the Gaussian Relay Channel with Energy Harvesting Constraints
Venue: arXiv cs.IT 2012
Year: 2012

This paper considers the use of energy harvesters, instead of conventional time-invariant energy sources, in wireless cooperative communication. For the purpose of exposition, we study the classic three-node Gaussian relay channel with decode-and-forward (DF) relaying, in which the source and relay nodes transmit with power drawn from energy-harvesting (EH) sources. Assuming a deterministic EH model under which the energy arrival time and the harvested amount are known prior to transmission, the throughput maximization problem over a finite horizon of $N$ transmission blocks is investigated. In particular, two types of data traffic with different delay constraints are considered: delay-constrained (DC) traffic (for which only one-block decoding delay is allowed at the destination) and no-delay-constrained (NDC) traffic (for which arbitrary decoding delay up to $N$ blocks is allowed). For the DC case, we show that the joint source and relay power allocation over time is necessary to achieve the maximum throughput, and propose an efficient algorithm to compute the optimal power profiles. For the NDC case, although the throughput maximization problem is non-convex, we prove the optimality of a separation principle for the source and relay power allocation problems, based upon which a two-stage power allocation algorithm is developed to obtain the optimal source and relay power profiles separately. Furthermore, we compare the DC and NDC cases, and obtain the sufficient and necessary conditions under which the NDC case performs strictly better than the DC case. It is shown that NDC transmission is able to exploit a new form of diversity arising from the independent source and relay energy availability over time in cooperative communication, termed "energy diversity", even with time-invariant channels.
Source Coding When the Side Information May Be Delayed
Venue: arXiv cs.IT 2012
Year: 2012

For memoryless sources, delayed side information at the decoder does not improve the rate-distortion function. However, this is not the case for more general sources with memory, as demonstrated by a number of works focusing on the special case of (delayed) feedforward. In this paper, a setting is studied in which the encoder is potentially uncertain about the delay with which measurements of the side information are acquired at the decoder. Assuming a hidden Markov model for the sources, at first, a single-letter characterization is given for the set-up where the side information delay is arbitrary and known at the encoder, and the reconstruction at the destination is required to be (near) lossless. Then, with delay equal to zero or one source symbol, a single-letter characterization is given of the rate-distortion region for the case where side information may be delayed or not, unbeknownst to the encoder. The characterization is further extended to allow for additional information to be sent when the side information is not delayed. Finally, examples for binary and Gaussian sources are provided.