arXiv subjects

Vincent Y. F. Tan

Publications and source records attributed to Vincent Y. F. Tan.

At least 181 records · Page 10Linked to original sources

arXiv2017

Zero-Error Capacity of $P$-ary Shift Channels and FIFO Queues

The objects of study of this paper are communication channels in which the dominant type of noise are symbol shifts, the main motivating examples being timing and bit-shift channels. Two channel models are introduced and their zero-error capacities and zero-error-detection capacities determined by explicit constructions of optimal codes. Model A can be informally described as follows: 1) The information is stored in an $ n $-cell register, where each cell is either empty or contains a particle of one of $ P $ possible types, and 2) due to the imperfections of the device each of the particles may be shifted several cells away from its original position over time. Model B is an abstraction of a single-server queue: 1) The transmitter sends packets from a $ P $-ary alphabet through a queuing system with an infinite buffer and a First-In-First-Out (FIFO) service procedure, and 2) each packet is being processed by the server for a random number of time slots. More general models including additional types of noise that the particles/packets can experience are also studied, as are the continuous-time versions of these problems.

cs.IT↗

arXiv2017

On Achievable Rates of AWGN Energy-Harvesting Channels with Block Energy Arrival and Non-Vanishing Error Probabilities

This paper investigates the achievable rates of an additive white Gaussian noise (AWGN) energy-harvesting (EH) channel with an infinite battery. The EH process is characterized by a sequence of blocks of harvested energy, which is known causally at the source. The harvested energy remains constant within a block while the harvested energy across different blocks is characterized by a sequence of independent and identically distributed (i.i.d.) random variables. The blocks have length $L$, which can be interpreted as the coherence time of the energy arrival process. If $L$ is a constant or grows sublinearly in the blocklength $n$, we fully characterize the first-order term in the asymptotic expansion of the maximum transmission rate subject to a fixed tolerable error probability $\varepsilon$. The first-order term is known as the $\varepsilon$-capacity. In addition, we obtain lower and upper bounds on the second-order term in the asymptotic expansion, which reveal that the second order term scales as $\sqrt{\frac{L}{n}}$ for any $\varepsilon$ less than $1/2$. The lower bound is obtained through analyzing the save-and-transmit strategy. If $L$ grows linearly in $n$, we obtain lower and upper bounds on the $\varepsilon$-capacity, which coincide whenever the cumulative distribution function (cdf) of the EH random variable is continuous and strictly increasing. In order to achieve the lower bound, we have proposed a novel adaptive save-and-transmit strategy, which chooses different save-and-transmit codes across different blocks according to the energy variation across the blocks.

cs.IT↗

arXiv2017

Moderate deviation analysis for classical communication over quantum channels

We analyse families of codes for classical data transmission over quantum channels that have both a vanishing probability of error and a code rate approaching capacity as the code length increases. To characterise the fundamental tradeoff between decoding error, code rate and code length for such codes we introduce a quantum generalisation of the moderate deviation analysis proposed by Altug and Wagner as well as Polyanskiy and Verdu. We derive such a tradeoff for classical-quantum (as well as image-additive) channels in terms of the channel capacity and the channel dispersion, giving further evidence that the latter quantity characterises the necessary backoff from capacity when transmitting finite blocks of classical data. To derive these results we also study asymmetric binary quantum hypothesis testing in the moderate deviations regime. Due to the central importance of the latter task, we expect that our techniques will find further applications in the analysis of other quantum information processing tasks.

quant-ph↗

arXiv2017

A Tight Upper Bound on the Second-Order Coding Rate of the Parallel Gaussian Channel with Feedback

This paper investigates the asymptotic expansion for the maximum rate of fixed-length codes over a parallel Gaussian channel with feedback under the following setting: A peak power constraint is imposed on every transmitted codeword, and the average error probability of decoding the transmitted message is non-vanishing as the blocklength increases. It is well known that the presence of feedback does not increase the first-order asymptotics of the channel, i.e., capacity, in the asymptotic expansion, and the closed-form expression of the capacity can be obtained by the well-known water-filling algorithm. The main contribution of this paper is a self-contained proof of an upper bound on the second-order asymptotics of the parallel Gaussian channel with feedback. The proof techniques involve developing an information spectrum bound followed by using Curtiss' theorem to show that a sum of dependent random variables associated with the information spectrum bound converges in distribution to a sum of independent random variables, thus facilitating the use of the usual central limit theorem. Combined with existing achievability results, our result implies that the presence of feedback does not improve the second-order asymptotics.

cs.IT↗

arXiv2017

A Proof of the Strong Converse Theorem for Gaussian Broadcast Channels via the Gaussian Poincaré Inequality

We prove that the Gaussian broadcast channel with two destinations admits the strong converse property. This implies that for every sequence of block codes operated at a common rate pair with an asymptotic average error probability $<1$, the rate pair must lie within the capacity region derived by Cover and Bergmans. The main mathematical tool required for our analysis is a logarithmic Sobolev inequality known as the Gaussian Poincaré inequality.

cs.IT↗

arXiv2017

Scaling Exponent and Moderate Deviations Asymptotics of Polar Codes for the AWGN Channel

This paper investigates polar codes for the additive white Gaussian noise (AWGN) channel. The scaling exponent $μ$ of polar codes for a memoryless channel $q_{Y|X}$ with capacity $I(q_{Y|X})$ characterizes the closest gap between the capacity and non-asymptotic achievable rates in the following way: For a fixed $\varepsilon \in (0, 1)$, the gap between the capacity $I(q_{Y|X})$ and the maximum non-asymptotic rate $R_n^*$ achieved by a length-$n$ polar code with average error probability $\varepsilon$ scales as $n^{-1/μ}$, i.e., $I(q_{Y|X})-R_n^* = Θ(n^{-1/μ})$. It is well known that the scaling exponent $μ$ for any binary-input memoryless channel (BMC) with $I(q_{Y|X})\in(0,1)$ is bounded above by $4.714$, which was shown by an explicit construction of polar codes. Our main result shows that $4.714$ remains to be a valid upper bound on the scaling exponent for the AWGN channel. Our proof technique involves the following two ideas: (i) The capacity of the AWGN channel can be achieved within a gap of $O(n^{-1/μ}\sqrt{\log n})$ by using an input alphabet consisting of $n$ constellations and restricting the input distribution to be uniform; (ii) The capacity of a multiple access channel (MAC) with an input alphabet consisting of $n$ constellations can be achieved within a gap of $O(n^{-1/μ}\log n)$ by using a superposition of $\log n$ binary-input polar codes. In addition, we investigate the performance of polar codes in the moderate deviations regime where both the gap to capacity and the error probability vanish as $n$ grows. An explicit construction of polar codes is proposed to obey a certain tradeoff between the gap to capacity and the decay rate of the error probability for the AWGN channel.

cs.IT↗

arXiv2017

A Unified Convergence Analysis of the Multiplicative Update Algorithm for Regularized Nonnegative Matrix Factorization

The multiplicative update (MU) algorithm has been extensively used to estimate the basis and coefficient matrices in nonnegative matrix factorization (NMF) problems under a wide range of divergences and regularizers. However, theoretical convergence guarantees have only been derived for a few special divergences without regularization. In this work, we provide a conceptually simple, self-contained, and unified proof for the convergence of the MU algorithm applied on NMF with a wide range of divergences and regularizers. Our main result shows the sequence of iterates (i.e., pairs of basis and coefficient matrices) produced by the MU algorithm converges to the set of stationary points of the non-convex NMF optimization problem. Our proof strategy has the potential to open up new avenues for analyzing similar problems in machine learning and signal processing.

math.OC↗

arXiv2017

Rank-One NMF-Based Initialization for NMF and Relative Error Bounds under a Geometric Assumption

We propose a geometric assumption on nonnegative data matrices such that under this assumption, we are able to provide upper bounds (both deterministic and probabilistic) on the relative error of nonnegative matrix factorization (NMF). The algorithm we propose first uses the geometric assumption to obtain an exact clustering of the columns of the data matrix; subsequently, it employs several rank-one NMFs to obtain the final decomposition. When applied to data matrices generated from our statistical model, we observe that our proposed algorithm produces factor matrices with comparable relative errors vis-à-vis classical NMF algorithms but with much faster speeds. On face image and hyperspectral imaging datasets, we demonstrate that our algorithm provides an excellent initialization for applying other NMF algorithms at a low computational cost. Finally, we show on face and text datasets that the combinations of our algorithm and several classical NMF algorithms outperform other algorithms in terms of clustering performance.

stat.ML↗

arXiv2017

Hypothesis Testing under Mutual Information Privacy Constraints in the High Privacy Regime

Hypothesis testing is a statistical inference framework for determining the true distribution among a set of possible distributions for a given dataset. Privacy restrictions may require the curator of the data or the respondents themselves to share data with the test only after applying a randomizing privacy mechanism. This work considers mutual information (MI) as the privacy metric for measuring leakage. In addition, motivated by the Chernoff-Stein lemma, the relative entropy between pairs of distributions of the output (generated by the privacy mechanism) is chosen as the utility metric. For these metrics, the goal is to find the optimal privacy-utility trade-off (PUT) and the corresponding optimal privacy mechanism for both binary and m-ary hypothesis testing. Focusing on the high privacy regime, Euclidean information-theoretic approximations of the binary and m-ary PUT problems are developed. The solutions for the approximation problems clarify that an MI-based privacy metric preserves the privacy of the source symbols in inverse proportion to their likelihoods.

cs.IT↗

arXiv2017

Hypothesis Testing under Maximal Leakage Privacy Constraints

The problem of publishing privacy-guaranteed data for hypothesis testing is studied using the maximal leakage (ML) as a metric for privacy and the type-II error exponent as the utility metric. The optimal mechanism (random mapping) that maximizes utility for a bounded leakage guarantee is determined for the entire leakage range for binary datasets. For non-binary datasets, approximations in the high privacy and high utility regimes are developed. The results show that, for any desired leakage level, maximizing utility forces the ML privacy mechanism to reveal partial to complete knowledge about a subset of the source alphabet. The results developed on maximizing a convex function over a polytope may also of an independent interest.

cs.IT↗

arXiv2017

On the Reliability Function of the Common-Message Broadcast Channel with Variable-Length Feedback

We derive upper and lower bounds on the reliability function for the common-message discrete memoryless broadcast channel with variable-length feedback. We show that the bounds are tight when the broadcast channel is stochastically degraded. For the achievability part, we adapt Yamamoto and Itoh's coding scheme by controlling the expectation of the maximum of a set of stopping times. For the converse part, we adapt Burnashev's proof techniques for establishing the reliability functions for (point-to-point) discrete memoryless channels with variable-length feedback and sequential hypothesis testing.

cs.IT↗

arXiv2017

Polar Coding for the Binary Erasure Channel with Deletions

We study the application of polar codes in deletion channels by analyzing the cascade of a binary erasure channel (BEC) and a deletion channel. We show how polar codes can be used effectively on a BEC with a single deletion, and propose a list decoding algorithm with a cyclic redundancy check for this case. The decoding complexity is $O(N^2\log N)$, where $N$ is the blocklength of the code. An important contribution is an optimization of the amount of redundancy added to minimize the overall error probability. Our theoretical results are corroborated by numerical simulations which show that the list size can be reduced to one and the original message can be recovered with high probability as the length of the code grows.

cs.IT↗

arXiv2017

Discrete Lossy Gray-Wyner Revisited: Second-Order Asymptotics, Large and Moderate Deviations

In this paper, we revisit the discrete lossy Gray-Wyner problem. In particular, we derive its optimal second-order coding rate region, its error exponent (reliability function) and its moderate deviations constant under mild conditions on the source. To obtain the second-order asymptotics, we extend some ideas from Watanabe's work (2015). In particular, we leverage the properties of an appropriate generalization of the conditional distortion-tilted information density, which was first introduced by Kostina and Verdú (2012). The converse part uses a perturbation argument by Gu and Effros (2009) in their strong converse proof of the discrete Gray-Wyner problem. The achievability part uses two novel elements: (i) a generalization of various type covering lemmas; and (ii) the uniform continuity of the conditional rate-distortion function in both the source (joint) distribution and the distortion level. To obtain the error exponent, for the achievability part, we use the same generalized type covering lemma and for the converse, we use the strong converse together with a change-of-measure technique. Finally, to obtain the moderate deviations constant, we apply the moderate deviations theorem to probabilities defined in terms of information spectrum quantities.

cs.IT↗

arXiv2016

Empirical Output Distribution of Good Delay-Limited Codes for Quasi-Static Fading Channels

This paper considers delay-limited communication over quasi-static fading channels under a long-term power constraint. A sequence of length-$n$ delay-limited codes for a quasi-static fading channel is said to be capacity-achieving if the codes achieve the delay-limited capacity, which is defined to be the maximum rate achievable by delay-limited codes. The delay-limited capacity is sometimes referred to as the zero-outage capacity in wireless communications. The delay-limited capacity is the appropriate choice of performance measure for delay-sensitive applications such as voice and video over fading channels. It is shown that for any sequence of capacity-achieving delay-limited codes with vanishing error probabilities, the normalized relative entropy between the output distribution induced by the length-$n$ code and the $n$-fold product of the capacity-achieving output distribution, denoted by $\frac{1}{n}D(p_{Y^n}\|p_{Y^n}^*)$, converges to zero. Additionally, we extend our convergence result to capacity-achieving delay-limited codes with non-vanishing error probabilities.

cs.IT↗

arXiv2016

On the Scaling Exponent of Polar Codes for Binary-Input Energy-Harvesting Channels

This paper investigates the scaling exponent of polar codes for binary-input energy-harvesting (EH) channels with infinite-capacity batteries. The EH process is characterized by a sequence of i.i.d. random variables with finite variances. The scaling exponent $μ$ of polar codes for a binary-input memoryless channel (BMC) characterizes the closest gap between the capacity and non-asymptotic rates achieved by polar codes with error probabilities no larger than some non-vanishing $\varepsilon\in(0,1)$. It has been shown that for any $\varepsilon\in(0,1)$, the scaling exponent $μ$ for any binary-input memoryless symmetric channel (BMSC) with $I(q_{Y|X})\in(0,1)$ lies between 3.579 and 4.714 , where the upper bound $4.714$ was shown by an explicit construction of polar codes. Our main result shows that $4.714$ remains to be a valid upper bound on the scaling exponent for any binary-input EH channel, i.e., a BMC subject to additional EH constraints. Our result thus implies that the EH constraints do not worsen the rate of convergence to capacity if polar codes are employed. The main result is proved by leveraging the following three existing results: scaling exponent analyses for BMSCs, construction of polar codes designed for binary-input memoryless asymmetric channels, and the save-and-transmit strategy for EH channels.

cs.IT↗

arXiv2016

The Dispersion of Nearest-Neighbor Decoding for Additive Non-Gaussian Channels

We study the second-order asymptotics of information transmission using random Gaussian codebooks and nearest neighbor (NN) decoding over a power-limited stationary memoryless additive non-Gaussian noise channel. We show that the dispersion term depends on the non-Gaussian noise only through its second and fourth moments, thus complementing the capacity result (Lapidoth, 1996), which depends only on the second moment. Furthermore, we characterize the second-order asymptotics of point-to-point codes over $K$-sender interference networks with non-Gaussian additive noise. Specifically, we assume that each user's codebook is Gaussian and that NN decoding is employed, i.e., that interference from the $K-1$ unintended users (Gaussian interfering signals) is treated as noise at each decoder. We show that while the first-order term in the asymptotic expansion of the maximum number of messages depends on the power of the interferring codewords only through their sum, this does not hold for the second-order term.

cs.IT↗

arXiv2016

Achievable Rates for Gaussian Degraded Relay Channels with Non-Vanishing Error Probabilities

This paper revisits the Gaussian degraded relay channel, where the link that carries information from the source to the destination is a physically degraded version of the link that carries information from the source to the relay. The source and the relay are subject to expected power constraints. The $\varepsilon$-capacity of the channel is characterized and it is strictly larger than the capacity for $\varepsilon>0$, which implies that the channel does not possess the strong converse property. The proof of the achievability part is based on several key ideas: block Markov coding which is used in the classical decode-forward strategy, power control for Gaussian channels under expected power constraints, and a careful scaling between the block size and the total number of block uses. The converse part is proved by first establishing two non-asymptotic lower bounds on the error probability, which are derived from the type-II errors of some binary hypothesis tests. Subsequently, each lower bound is simplified by conditioning on an event related to the power of some linear combination of the codewords transmitted by the source and the relay. Lower and upper bounds on the second-order term of the optimal coding rate in terms of blocklength and error probability are also obtained.

cs.IT↗

arXiv2016

Online Nonnegative Matrix Factorization with Outliers

We propose a unified and systematic framework for performing online nonnegative matrix factorization in the presence of outliers. Our framework is particularly suited to large-scale data. We propose two solvers based on projected gradient descent and the alternating direction method of multipliers. We prove that the sequence of objective values converges almost surely by appealing to the quasi-martingale convergence theorem. We also show the sequence of learned dictionaries converges to the set of stationary points of the expected loss function almost surely. In addition, we extend our basic problem formulation to various settings with different constraints and regularizers. We also adapt the solvers and analyses to each setting. We perform extensive experiments on both synthetic and real datasets. These experiments demonstrate the computational efficiency and efficacy of our algorithms on tasks such as (parts-based) basis learning, image denoising, shadow removal and foreground-background separation.

stat.ML↗