The price of ignorance: The impact of side-information on delay for lossless source-coding

Inspired by the context of compressing encrypted sources, this paper considers the general tradeoff between rate, end-to-end delay, and probability of error for lossless source coding with side-information. The notion of end-to-end delay is made precise by considering a sequential setting in which source symbols are revealed in real time and need to be reconstructed at the decoder within a certain fixed latency requirement. Upper bounds are derived on the reliability functions with delay when side-information is known only to the decoder as well as when it is also known at the encoder. When the encoder is not ignorant of the side-information (including the trivial case when there is no side-information), it is possible to have substantially better tradeoffs between delay and probability of error at all rates. This shows that there is a fundamental price of ignorance in terms of end-to-end delay when the encoder is not aware of the side information. This effect is not visible if only fixed-block-length codes are considered. In this way, side-information in source-coding plays a role analogous to that of feedback in channel coding. While the theorems in this paper are asymptotic in terms of long delays and low probabilities of error, an example is used to show that the qualitative effects described here are significant even at short and moderate delays.Comment: 25 pages, 17 figures. Submitted to the IEEE Transactions on Information Theor

Optimal Lempel-Ziv based lossy compression for memoryless data: how to make the right mistakes

Compression refers to encoding data using bits, so that the representation uses as few bits as possible. Compression could be lossless: i.e. encoded data can be recovered exactly from its representation) or lossy where the data is compressed more than the lossless case, but can still be recovered to within prespecified distortion metric. In this paper, we prove the optimality of Codelet Parsing, a quasi-linear time algorithm for lossy compression of sequences of bits that are independently and identically distributed (\iid) and Hamming distortion. Codelet Parsing extends the lossless Lempel Ziv algorithm to the lossy case---a task that has been a focus of the source coding literature for better part of two decades now. Given \iid sequences \x, the expected length of the shortest lossy representation such that \x can be reconstructed to within distortion \dist is given by the rate distortion function, \rd. We prove the optimality of the Codelet Parsing algorithm for lossy compression of memoryless bit sequences. It splits the input sequence naturally into phrases, representing each phrase by a codelet, a potentially distorted phrase of the same length. The codelets in the lossy representation of a length-$n$ string {\x} have length roughly (\log n)/\rd, and like the lossless Lempel Ziv algorithm, Codelet Parsing constructs codebooks logarithmic in the sequence length.Comment: This file is not the final version, and will be updated for the next few days. (Edited 10/17

One-pass adaptive universal vector quantization

The authors introduce a one-pass adaptive universal quantization technique for real, bounded alphabet, stationary sources. The algorithm is set on line without any prior knowledge of the statistics of the sources which it might encounter and asymptotically achieves ideal performance on all sources that it sees. The system consists of an encoder and a decoder. At increasing intervals, the encoder refines its codebook using knowledge about incoming data symbols. This codebook is then described to the decoder in the form of updates on the previous codebook. The accuracy to which the codebook is described increases as the number of symbols seen, and thus the accuracy to which the codebook is known, grows

The Shannon Cipher System with a Guessing Wiretapper: General Sources

The Shannon cipher system is studied in the context of general sources using a notion of computational secrecy introduced by Merhav & Arikan. Bounds are derived on limiting exponents of guessing moments for general sources. The bounds are shown to be tight for iid, Markov, and unifilar sources, thus recovering some known results. A close relationship between error exponents and correct decoding exponents for fixed rate source compression on the one hand and exponents for guessing moments on the other hand is established.Comment: 24 pages, Submitted to IEEE Transactions on Information Theor

Sequential Source Coding for Stochastic Systems Subject to Finite Rate Constraints

In this paper, we revisit the sequential source coding framework to analyze fundamental performance limitations of discrete-time stochastic control systems subject to feedback data-rate constraints in finite-time horizon. The basis of our results is a new characterization of the lower bound on the minimum total-rate achieved by sequential codes subject to a total (across time) distortion constraint and a computational algorithm that allocates optimally the rate-distortion for any fixed finite-time horizon. This characterization facilitates the derivation of analytical, non-asymptotic, and finite-dimensional lower and upper bounds in two control-related scenarios. (a) A parallel time-varying Gauss-Markov process with identically distributed spatial components that is quantized and transmitted through a noiseless channel to a minimum mean-squared error (MMSE) decoder. (b) A time-varying quantized LQG closed-loop control system, with identically distributed spatial components and with a random data-rate allocation. Our non-asymptotic lower bound on the quantized LQG control problem, reveals the absolute minimum data-rates for (mean square) stability of our time-varying plant for any fixed finite time horizon. We supplement our framework with illustrative simulation experiments.Comment: 40 pages, 6 figure

Interactive Communication for Data Exchange

Two parties observing correlated data seek to exchange their data using interactive communication. How many bits must they communicate? We propose a new interactive protocol for data exchange which increases the communication size in steps until the task is done. Next, we derive a lower bound on the minimum number of bits that is based on relating the data exchange problem to the secret key agreement problem. Our single-shot analysis applies to all discrete random variables and yields upper and lower bound of a similar form. In fact, the bounds are asymptotically tight and lead to a characterization of the optimal rate of communication needed for data exchange for a general sequence such as mixture of IID random variables as well as the optimal second-order asymptotic term in the length of communication needed for data exchange for the IID random variables, when the probability of error is fixed. This gives a precise characterization of the asymptotic reduction in the length of optimal communication due to interaction; in particular, two-sided Slepian-Wolf compression is strictly suboptimal.Comment: 13 pages (two column), 4 figures, a longer version of ISIT 201

Robust Bayesian compressed sensing over finite fields: asymptotic performance analysis

This paper addresses the topic of robust Bayesian compressed sensing over finite fields. For stationary and ergodic sources, it provides asymptotic (with the size of the vector to estimate) necessary and sufficient conditions on the number of required measurements to achieve vanishing reconstruction error, in presence of sensing and communication noise. In all considered cases, the necessary and sufficient conditions asymptotically coincide. Conditions on the sparsity of the sensing matrix are established in presence of communication noise. Several previously published results are generalized and extended.Comment: 42 pages, 4 figure

Distributed Joint Source-Channel Coding on a Multiple Access Channel with Side Information

We consider the problem of transmission of several distributed sources over a multiple access channel (MAC) with side information at the sources and the decoder. Source-channel separation does not hold for this channel. Sufficient conditions are provided for transmission of sources with a given distortion. The source and/or the channel could have continuous alphabets (thus Gaussian sources and Gaussian MACs are special cases). Various previous results are obtained as special cases. We also provide several good joint source-channel coding schemes for a discrete/continuous source and discrete/continuous alphabet channel. Channels with feedback and fading are also considered. Keywords: Multiple access channel, side information, lossy joint source-channel coding, channels with feedback, fading channels.Comment: 49 pages, Technical Report, DRDO-IISc programme on Advanced Research in Mathematical Engineering, Dept of ECE, Indian Institute of Science, Bangalore, Indi

EγE_{\gamma}-Resolvability

The conventional channel resolvability refers to the minimum rate needed for an input process to approximate the channel output distribution in total variation distance. In this paper we study $E_{\gamma}$-resolvability, in which total variation is replaced by the more general $E_{\gamma}$ distance. A general one-shot achievability bound for the precision of such an approximation is developed. Let $Q_{\sf X|U}$ be a random transformation, $n$ be an integer, and $E\in(0,+\infty)$. We show that in the asymptotic setting where $\gamma=\exp(nE)$, a (nonnegative) randomness rate above $\inf_{Q_{\sf U}: D(Q_{\sf X}\|{{\pi}}_{\sf X})\le E} \{D(Q_{\sf X}\|{{\pi}}_{\sf X})+I(Q_{\sf U},Q_{\sf X|U})-E\}$ is sufficient to approximate the output distribution ${{\pi}}_{\sf X}^{\otimes n}$ using the channel $Q_{\sf X|U}^{\otimes n}$, where $Q_{\sf U}\to Q_{\sf X|U}\to Q_{\sf X}$, and is also necessary in the case of finite $\mathcal{U}$ and $\mathcal{X}$. In particular, a randomness rate of $\inf_{Q_{\sf U}}I(Q_{\sf U},Q_{\sf X|U})-E$ is always sufficient. We also study the convergence of the approximation error under the high probability criteria in the case of random codebooks. Moreover, by developing simple bounds relating $E_{\gamma}$ and other distance measures, we are able to determine the exact linear growth rate of the approximation errors measured in relative entropy and smooth R\'{e}nyi divergences for a fixed-input randomness rate. The new resolvability result is then used to derive 1) a one-shot upper bound on the probability of excess distortion in lossy compression, which is exponentially tight in the i.i.d.~setting, 2) a one-shot version of the mutual covering lemma, and 3) a lower bound on the size of the eavesdropper list to include the actual message and a lower bound on the eavesdropper false-alarm probability in the wiretap channel problem, which is (asymptotically) ensemble-tight.Comment: 30 pages, 5 figures, presented in part at 2015 IEEE International Symposium on Information Theory (ISIT

The Rate-Distortion Risk in Estimation from Compressed Data

Consider the problem of estimating a latent signal from a lossy compressed version of the data. Assume that the data is compressed to a prescribed bitrate via a procedure that is agnostic to the model describing the relation between the latent signal and the data. In reconstruction, the latent signal is estimated to minimize a prescribed risk function. For the above setting and a given distortion measure between the data and its compressed version, we define the rate-distortion (RD) risk of an estimator as its risk under the distribution achieving Shannon's RD function at the prescribed bitrate. We derive conditions on the compression code under which the true risk in estimating from the compressed data is asymptotically equivalent to the RD risk. The main theoretical tools to obtain these conditions are transportation-cost inequalities in conjunction with properties of compression codes achieving Shannon's RD function. We show that these conditions typically hold in a memoryless discrete setting or when the RD achieving distribution is multivariate normal. Whenever the aforementioned asymptotic equivalence holds, this RD risk provides an achievable estimation performance in situations when the data is compressed, communicated, or stored using a procedure that is agnostic to the latent signal or the ultimate inference task. Furthermore, in these cases, our results imply a general procedure for designing an estimator from a dataset undergoing lossy compression without specifying the actual compression technique. Namely, by designing it based on the RD achieving distribution.Comment: Under review for the IEEE Transactions on Information Theor