Error Exponent Research Articles

Quantized CP approximation and sparse tensor interpolation of function‐generated data

SummaryIn this article, we consider the iterative schemes to compute the canonical polyadic (CP) approximation of quantized data generated by a function discretized on a large uniform grid in an interval on the real line. This paper continues the research on the quantics‐tensor train (QTT) method (“O(d log N)‐quantics approximation of N‐d tensors in high‐dimensional numerical modeling” in Constructive Approximation, 2011) developed for the tensor train (TT) approximation of the quantized images of function related data. In the QTT approach, the target vector of length 2L is reshaped to a Lth‐order tensor with two entries in each mode (quantized representation) and then approximated by the QTT tensor including 2r2L parameters, where r is the maximal TT rank. In what follows, we consider the alternating least squares (ALS) iterative scheme to compute the rank‐r CP approximation of the quantized vectors, which requires only 2rL≪2L parameters for storage. In the earlier papers (“Tensors‐structured numerical methods in scientific computing: survey on recent advances” in Chemom Intell Lab Syst, 2012), such a representation was called QCan format, whereas in this paper, we abbreviate it as the QCP (quantized canonical polyadic) representation. We test the ALS algorithm to calculate the QCP approximation on various functions, and in all cases, we observed the exponential error decay in the QCP rank. The main idea for recovering a discretized function in the rank‐r QCP format using the reduced number of the functional samples, calculated only at O(2rL) grid points, is presented. The special version of the ALS scheme for solving the arising minimization problem is described. This approach can be viewed as the sparse QCP‐interpolation method that allows to recover all 2rL representation parameters of the rank‐r QCP tensor. Numerical examples show the efficiency of the QCP‐ALS‐type iteration and indicate the exponential convergence rate in r.

Time‐dependent radiation pattern analysis of cable‐net reflector antennas

Due to the time-dependent load-elongation characteristic of cables, the cable-net reflector antenna shows a creep and recovery behaviour which has limited the development of the cable-net reflector antenna with high frequency and high stability. Therefore, this study is dedicated to investigating a time-dependent radiation pattern analysis method for cable-net reflector antennas. Based on the physical optics (PO) method, formulas are firstly established for the time-dependent far-field radiated pattern analysis. To avoid the repetitive computation of PO radiation integrals and improve the calculation efficiency, the exponential error terms are expanded by the Taylor series at the mean values of phase errors in each small region and the time factor is separated from the integral for some special case. Eventually, an axis-symmetric reflector antenna is taken as a numerical example to verify the practicability, validity and robustness of the proposed method.

Capacity-Achieving Guessing Random Additive Noise Decoding

We introduce a new algorithm for realizing Maximum Likelihood (ML) decoding in discrete channels with or without memory. In it, the receiver rank orders noise sequences from most likely to least likely. Subtracting noise from the received signal in that order, the first instance that results in a member of the code-book is the ML decoding. We name this algorithm GRAND for Guessing Random Additive Noise Decoding. We establish that GRAND is capacity-achieving when used with random code-books. For rates below capacity we identify error exponents, and for rates beyond capacity we identify success exponents. We determine the scheme's complexity in terms of the number of computations the receiver performs. For rates beyond capacity, this reveals thresholds for the number of guesses by which if a member of the code-book is identified it is likely to be the transmitted code-word. We introduce an approximate ML decoding scheme where the receiver abandons the search after a fixed number of queries, an approach we dub GRANDAB, for GRAND with ABandonment. While not an ML decoder, we establish that the algorithm GRANDAB is also capacity-achieving for an appropriate choice of abandonment threshold, and characterize its complexity, error and success exponents. Worked examples are presented for Markovian noise that indicate these decoding schemes substantially out-perform the brute force decoding approach.

Hypothesis Testing Over the Two-Hop Relay Network

Coding and testing schemes and the corresponding achievable type-II error exponents are presented for binary hypothesis testing over two-hop relay networks. The schemes are based on cascade source coding techniques and {unanimous decision-forwarding}, the latter meaning that a terminal decides on the null hypothesis only if all previous terminals have decided on the null hypothesis. If the observations at the transmitter, the relay, and the receiver form a Markov chain in this order, then, without loss in performance, the proposed cascade source code can be replaced by two independent point-to-point source codes, one for each hop. The decoupled scheme (combined with decision-forwarding) is shown to attain the optimal type-II error exponents for various instances of "testing against conditional independence." The same decoupling is shown to be optimal also for some instances of "testing against independence," when the observations at the transmitter, the receiver, and the relay form a Markov chain in this order, and when the relay-to-receiver link is of sufficiently high rate. For completeness, the paper also presents an analysis of the Shimokawa-Han-Amari binning scheme for the point-to-point hypothesis testing setup.

Exact Error and Erasure Exponents for the Asymmetric Broadcast Channel

Consider the asymmetric broadcast channel with a random superposition codebook, which may be comprised of constant composition or i.i.d. codewords. By applying Forney’s optimal decoder for individual messages and the message pair for the receiver that decodes both messages, exact (ensemble-tight) error and erasure exponents are derived. It is shown that the optimal decoder designed to decode the pair of messages achieves the optimal trade-off between the total and undetected exponents associated with the optimal decoder for the private message. Convex optimization-based procedures to evaluate the exponents efficiently are proposed. Finally, numerical examples are presented to illustrate the results.

Speculate-correct error bounds for k-nearest neighbor classifiers

We introduce the speculate-correct method to derive error bounds for local classifiers. Using it, we show that k nearest neighbor classifiers, in spite of their famously fractured decision boundaries, have exponential error bounds with O(sqrt((k + ln n) / n)) error bound range for n in-sample examples.

Expurgated Bounds for the Asymmetric Broadcast Channel

This work contains two main contributions concerning the expurgation of hierarchical ensembles for the asymmetric broadcast channel. The first is an analysis of the optimal maximum likelihood (ML) decoders for the weak and strong user. Two different methods of code expurgation will be used, that will provide two competing error exponents. The second is the derivation of expurgated exponents under the generalized stochastic likelihood decoder (GLD). We prove that the GLD exponents are at least as tight as the maximum between the random coding error exponents derived in an earlier work by Averbuch and Merhav (2017) and one of our ML-based expurgated exponents. By that, we actually prove the existence of hierarchical codebooks that achieve the best of the random coding exponent and the expurgated exponent simultaneously for both users.

Generalized Random Gilbert-Varshamov Codes

We introduce a random coding technique for transmission over discrete memoryless channels, reminiscent of the basic construction attaining the Gilbert–Varshamov bound for codes in Hamming spaces. The code construction is based on drawing codewords recursively from a fixed type class, in such a way that a newly generated codeword must be at a certain minimum distance from all previously chosen codewords, according to some generic distance function. We derive an achievable error exponent for this construction and prove its tightness with respect to the ensemble average. We show that the exponent recovers the Csiszar and Korner exponent as a special case, which is known to be at least as high as both the random-coding and expurgated exponents, and we establish the optimality of certain choices of the distance function. In addition, for additive distances and decoding metrics, we present an equivalent dual expression, along with a generalization to infinite alphabets via cost-constrained random coding.

Molecules with dipoles in periodic boundary conditions in a tetragonal cell

When a system which contains a dipole, and whose dimensionality is less than three, is studied in a code which imposes periodic boundary conditions in all three dimensions, an artificial electric field arises which keeps the potential periodic. This has an impact on the total energy of the system, and on any other attribute which would respond to an electric field. Simple corrections are known for 0D systems embedded in a cubic geometry, and 2D slab systems. This paper shows how the 0D result can be extended to tetragonal geometries, and that for a particular c/a ratio the correction is zero. It also considers an exponential error term absent from the usual consideration of 2D slab geometries, and discusses an empirical form for this.

Error Exponents for Dimension-Matched Vector Multiple Access Channels With Additive Noise

We analyze a class of vector multiple access channels with additive noise, where the sum of the dimensions of the transmitted signals matches that of the received signal. We first focus on the case without power constraints, in the Poltyrev sense, using point process techniques. We find the Poltyrev capacity region for noise processes that are independent and identically distributed over channel uses. For each rate vector strictly in the Poltyrev capacity region, we study, for each subset of the transmitters, the exponent of the decay in block length of the smallest possible probability that decoding results in error for each transmitter in that subset. In the case of independent and identically distributed Gaussian noise, with arbitrary positive definite covariance matrix, we derive random coding exponents for each type of error event—these are lower bounds to the true error exponents. This also leads to random coding error exponents in the traditional power-constrained case, where the power constraint at each transmitter is defined by an arbitrary positive definite matrix.

Quantum Sphere-Packing Bounds With Polynomial Prefactors

We study lower bounds on the optimal error probability in classical coding over classical-quantum channels at rates below the capacity, commonly termed quantum sphere-packing bounds. Winter and Dalai have derived such bounds for classical-quantum channels; however, the exponents in their bounds only coincide when the channel is classical. In this paper, we show that these two exponents admit a variational representation and are related by the Golden-Thompson inequality, reaffirming that Dalai's expression is stronger in general classical-quantum channels. Second, we establish a sphere-packing bound for classical-quantum channels, which significantly improves Dalai's prefactor from the order of subexponential to polynomial. Furthermore, the gap between the obtained error exponent for constant composition codes and the best known classical random coding exponent vanishes in the order of $o(\log n / n)$, indicating our sphere-packing bound is almost exact in the high rate regime. Finally, for a special class of symmetric classical-quantum channels, we can completely characterize its optimal error probability without the constant composition code assumption. The main technical contributions are two converse Hoeffding bounds for quantum hypothesis testing and the saddle-point properties of error exponent functions.

The Error Probability of Sparse Superposition Codes With Approximate Message Passing Decoding

Sparse superposition codes, or sparse regression codes (SPARCs), are a recent class of codes for reliable communication over the AWGN channel at rates approaching the channel capacity. Approximate message passing (AMP) decoding, a computationally efficient technique for decoding SPARCs, has been proven to be asymptotically capacity-achieving for the AWGN channel. In this paper, we refine the asymptotic result by deriving a large deviations bound on the probability of AMP decoding error. This bound gives insight into the error performance of the AMP decoder for large but finite problem sizes, giving an error exponent as well as guidance on how the code parameters should be chosen at finite block lengths. For an appropriate choice of code parameters, we show that for any fixed rate less than the channel capacity, the decoding error probability decays exponentially in $n/(\log n)^{2T}$, where $T$, the number of AMP iterations required for successful decoding, is bounded in terms of the gap from capacity.

Adaptive Estimation of Time-Varying Parameters With Application to Roto-Magnet Plant

This paper presents an alternative adaptive parameter estimation framework for nonlinear systems with time-varying parameters. Unlike existing techniques that rely on the polynomial approximation of time-varying parameters, the proposed method can directly estimate the unknown time-varying parameters. Moreover, this paper proposes several new adaptive laws driven by the derived information of parameter estimation errors, which achieve faster convergence rate than conventional gradient descent algorithms. In particular, the exponential error convergence can be rigorously proved under the well-recognized persistent excitation condition. The robustness of the developed adaptive estimation schemes against bounded disturbances is also studied. Comparative simulation results reveal that the proposed approaches can achieve better estimation performance than several other estimation algorithms. Finally, the proposed parameter estimation methods are verified by conducting experiments based on a roto-magnet plant.

Output Feedback Control of Uncertain Hydraulic Servo Systems

Most of the existing control methods for servo systems driven by hydraulic actuators have been developed by using a backstepping scheme and assuming that all system states (including internal hydraulic signals) are measurable. In this paper, we propose a new control design method for high-order servo systems with hydraulic actuator dynamics, where the backstepping scheme is avoided and only the system output (e.g., motion displacement) is required for the control implementation. For this purpose, the system model is first transformed into a canonical form, where the unknown dynamics in the system are lumped as one term. Then, we introduce a simple robust unknown dynamics estimator (UDE) that has only one tuning parameter but achieves exponential error convergence to accommodate the lumped uncertainties. Therefore, the function approximators (e.g., neural network and fuzzy systems) can be avoided, leading to reduced computational costs, simpler parameter tuning, and improved convergence as compared to backstepping methods. Extensive simulations and experiments based on a realistic test rig are conducted to show the efficacy of the proposed control.

Performance of Viterbi Decoding With and Without ARQ on Rician Fading Channels

In this paper, we investigate the performance of the Viterbi decoding algorithm with/without Automatic Repeat reQuest (ARQ) over a Rician flat fading channel with unlimited interleaving. We show that the decay rate of the average bit error probability with respect to the bit energy to noise ratio is at least equal to d <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">f</sub> at high-bit energy to noise ratio for both cases (with ARQ and without ARQ), where d <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">f</sub> is the free distance of the convolutional code. The Yamamoto-Itoh flag helps to reduce the average bit error probability by a factor of 4(d) <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">f</sub> with a negligible retransmission rate. We also prove an interesting result that the average bit error probability decays exponentially fast with respect to the Rician factor for any fixed bit energy per noise ratio. In addition, the average bit error exponent with respect to the Rician factor is shown to be d <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">f</sub> .

Integral partition bounds for fast and effective screening of general one-, two-, and many-electron integrals.

We introduce tight upper bounds for a variety of integrals appearing in electronic structure theories. These include electronic interaction integrals involving any number of electrons and various integral kernels such as the ubiquitous electron repulsion integrals and the three- and four-electron integrals found in explicitly correlated methods. Our bounds are also applicable to the one-electron potential integrals that appear in great number in quantum mechanical (QM), mixed quantum and molecular mechanical (QM/MM), and semi-numerical methods. The bounds are based on a partitioning of the integration space into balls centered around electronic distributions and their complements. Such a partitioning leads directly to equations for rigorous extents, which we solve for shell pair distributions containing shells of Gaussian basis functions of arbitrary angular momentum. The extents are the first general rigorous formulation we are aware of, as previous definitions are based on the inverse distance operator 1/r12 and typically only rigorous for simple spherical Gaussians. We test our bounds for six different integral kernels found throughout quantum chemistry, including exponential, Gaussian, and complementary error function based forms. We compare to previously developed estimates on the basis of significant integral counts and their usage in both explicitly correlated second-order Møller-Plesset theory (MP2-F12) and density functional theory calculations employing screened Hartree-Fock exchange.

Universal Channel Coding for General Output Alphabet

We propose two types of universal codes that are suited to two asymptotic regimes when the output alphabet is possibly continuous. The first class has the property that the error probability decays exponentially fast and we identify an explicit lower bound on the error exponent. The other class attains the epsilon-capacity the channel and we also identify the second-order term in the asymptotic expansion. The proposed encoder is essentially based on the packing lemma of the method of types. For the decoder, we first derive a R\'enyi-relative-entropy version of Clarke and Barron's formula the distance between the true distribution and the Bayesian mixture, which is of independent interest. The universal decoder is stated in terms of this formula and quantities used in the information spectrum method. The methods contained herein allow us to analyze universal codes for channels with continuous and discrete output alphabets in a unified manner, and to analyze their performances in terms of the exponential decay of the error probability and the second-order coding rate.

AN EXPLICIT TWO‐SOURCE EXTRACTOR WITH MIN‐ENTROPY RATE NEAR

In 2005 Bourgain gave the first explicit construction of a two-source extractor family with min-entropy rate less than $1/2$. His approach combined Fourier analysis with innovative but inefficient tools from arithmetic combinatorics and yielded an unspecified min-entropy rate which was greater than $.499$. This remained essentially the state of the art until a 2015 breakthrough of Chattopadhyay and Zuckerman in which they gave an alternative approach which produced extractors with arbitrarily small min-entropy rate. In the current work, we revisit the Fourier analytic approach. We give an improved analysis of one of Bourgain's extractors which shows that it in fact extracts from sources with min-entropy rate near $\frac{21}{44} =.477\ldots$, moreover we construct a variant of this extractor which we show extracts from sources with min-entropy rate near $4/9 $ = $.444\ldots$. While this min-entropy rate is inferior to Chattopadhyay and Zuckerman's construction, our extractors have the advantage of exponential small error which is important in some applications. The key ingredient in these arguments is recent progress connected to the restriction theory of the finite field paraboloid by Rudnev and Shkredov. This in turn relies on a Rudnev's point-plane incidence estimate, which in turn relies on Koll\'ar's generalization of the Guth-Katz incidence theorem.

Exponential Observation Error in Boundary Region

In this paper the (exponential) perception blunder idea on account of limit locale has been discussed and broke down. For disseminated parameter frameworks of explanatory sort, we demonstrate that, the blunder of state reproduction can be diminishes by exponentially perception.

The problem of constructing unsaturated quadrature formulae on an interval

Unsaturated quadrature formulae are constructed which are well conditioned on the finite interval with -weight function, . A specific feature of such formulae is the absence of the principal error term, which ensures that they can be automatically readjusted (with an increased number of nodes) to any excessive (extraordinary) amount of smoothness of the integrands. All the key parameters of quadratures (the nodes, the coefficients and the condition number) are evaluated within a single general approach based on the solution of a number of special boundary-value problems in the theory of meromorphic functions in the unit disc. For particular weight functions, which have important applications, algorithms for evaluating all the parameters of the quadratures efficiently are put forward. For -smooth integrands, an answer is given with an absolutely sharp exponential error estimate. The sharpness of the estimate is secured by the asymptotic behaviour of the Alexandrov -width of a compact set of -smooth functions, which goes to zero exponentially (as the number of nodes goes off to infinity). Bibliography: 32 titles.