chebyshev hankel matrices and the splitting approach for centrosymmetric toeplitz plus hankel matrices abstract centrosymmetric toeplitz plus hankel matrices are investigated on the basis of their splitting property which is their similarity to the direct sum of two special toeplitz plus hankel matrices these matrices can be considered as hankel matrices moment matrices in bases of chebyshev polynomials and are called chebyshev hankel matrices chebyshev hankel matrices have similar properties like hankel matrices this concerns inversion formulas and fast algorithms a superfast algorithm for solving chebyshev hankel and centrosymmetric toeplitz plus hankel systems is presented that is based on real trigonometric transforms the main tool of investigation is the interpretation of chebyshev hankel matrices as matrices of restricted multiplication operators with respect to chebyshev bases

the levinson recurrence and fast algorithms for solving toeplitz systems of linear equations this work brings together classical polynomial theory as it relates to the levinson recurrence for a hermitian toeplitz operator and matrix theory as it relates to the class of hermitian centro hermitian matrices a new computationally efficient alternative is presented to the levinson recurrence on either the hermitian or skew hermitian polynomial spaces this approach also leads to an entirely new algorithm for solving systems of linear equations when the coefficient matrix is hermitian toeplitz or real symmetric toeplitz analysis of the computational complexity of the algorithms presented is also performed and it is shown that these algorithms lead to significant improvements in the computational complexity as compared to the previously best known recursive algorithms they also provide further insight into the mathematical properties of the structurally rich toeplitz matrices

the split levinson algorithm the classical levinson algorithm for computing the predictor polynomial relative to a real positive definite toeplitz matrix is shown to be redundant in complexity it can be broken down into two simpler algorithms either of which needs only to be processed this result can be interpreted in the framework of the theory of orthogonal polynomials on the real line as follows the symmetric and antisymmetric parts of the predictors relative to the sequence of toeplitz matrices constitute two families of polynomials orthogonal on the interval 1 1 with respect to some even weight functions it turns out that the recurrence relations for these orthogonal polynomials can be used efficiently to compute the desired predictor the resulting split levinson algorithm requires roughly one half the number of multiplications and the same number of additions as the classical levinson algorithm a simple derivation of cybenko s method for computing the pisarenko frequencies is obtained from the recurrence relations underlying the split levinson algorithm

stochastic thermodynamics with information reservoirs we generalize stochastic thermodynamics to include information reservoirs such information reservoirs which can be modeled as a sequence of bits modify the second law for example work extraction from a system in contact with a single heat bath becomes possible if the system also interacts with an information reservoir we obtain an inequality and the corresponding fluctuation theorem generalizing the standard entropy production of stochastic thermodynamics from this inequality we can derive an information processing entropy production which gives the second law in the presence of information reservoirs we also develop a systematic linear response theory for information processing machines for a unicyclic machine powered by an information reservoir the efficiency at maximum power can deviate from the standard value of 1 2 for the case where energy is consumed to erase the tape the efficiency at maximum erasure rate is found to be 1 2

a generalization of the levinson algorithm for hermitian toeplitz matrices with any rank profile the paper describes a recursive algorithm for solving hermitian toeplitz systems of linear equations without any restriction on the ranks of their nested toeplitz subsystems such a general algorithm is needed e g to obtain the eigenfilters for signal processing applications or to compute the inverse of a nondefinite toeplitz matrix the regular portion of the algorithm is made of the classical levinson recursion the singular portion requires solving some well defined systems of linear equations with gradient structure the dimension of each of these sytems equals the amplitude of the corresponding singularity

effect of multiplicative noise on stationary stochastic process an open system that can be analyzed using the langevin equation with multiplicative noise is considered the stationary state of the system results from a balance of deterministic damping and random pumping simulated as noise with controlled periodicity the dependence of statistical moments of the variable that characterizes the system on parameters of the problem is studied a nontrivial decrease in the mean value of the main variable with an increase in noise stochasticity is revealed applications of the results in several physical chemical biological and technical problems of natural and humanitarian sciences are discussed

on the splitting of classical algorithms in linear prediction theory the levinson algorithm has been shown recently to be redundant in complexity it can be broken down into two simpler algorithms either of these split levinson algorithms needs only to be processed in the present paper it is shown that the schur algorithm the lattice algorithm and the normalized lattice algorithm for computing reflection coefficients can be broken down in a similar manner the number of multiplications in each of the resulting split algorithms is reduced with respect to the classical version by a factor lying between 1 5 and 2

fluctuation theorem for currents and schnakenberg network theory a fluctuation theorem is proved for the macroscopic currents of a system in a nonequilibrium steady state by using schnakenberg network theory the theorem can be applied in particular in reaction systems where the affinities or thermodynamic forces are defined globally in terms of the cycles of the graph associated with the stochastic process describing the time evolution

parallel and superfast algorithms for hankel systems of equations utilizing kernel structure properties a unified construction of hankel matrix inversion algorithms is presented three types of algorithms are obtained 1 o n 2 complexity levinson type 2 o n parallel complexity schur type and 3 o n log2 n complexity asymptotically fast ones all algorithms work without additional assumption like strong nonsingularity

nonequilibrium detailed fluctuation theorem for repeated discrete feedback we extend the framework of forward and reverse processes commonly utilized in the derivation and analysis of the nonequilibrium work relations to thermodynamic processes with repeated discrete feedback within this framework we derive a generalization of the detailed fluctuation theorem which is modified by the addition of a term that quantifies the change in uncertainty about the microscopic state of the system upon making measurements of physical observables during feedback as an application we extend two nonequilibrium work relations the nonequilibrium work fluctuation theorem and the relative entropy work relation

minimal energy cost for thermodynamic information processing measurement and information erasure the fundamental lower bounds on the thermodynamic energy cost of measurement and information erasure are determined the lower bound on the erasure validates landauer s principle for a symmetric memory for other cases the bound indicates the breakdown of the principle our results constitute the second law of information thermodynamics in which information content and thermodynamic variables are treated on an equal footing

thermodynamics of genuine nonequilibrium states under feedback control for genuine nonequilibrium states that even at fixed external control parameter exhibit dissipation we extend the hatano sasa equality to processes with feedback control the resulting bound on the maximal extractable work is substantially sharper than what would follow from applying the sagawa ueda equality to transitions involving such states for repeated measurements at short enough intervals the power thus extracted can even exceed the average cost of driving as demonstrated explicitly with a simple analytically solvable example

recursive nearest agglomeration rena fast clustering for approximation of structured signals in this work we revisit fast dimension reduction approaches as with random projections and random sampling our goal is to summarize the data to decrease computational costs and memory footprint of subsequent analysis such dimension reduction can be very efficient when the signals of interest have a strong structure such as with images we focus on this setting and investigate feature clustering schemes for data reductions that capture this structure an impediment to fast dimension reduction is then that good clustering comes with large algorithmic costs we address it by contributing a linear time agglomerative clustering scheme recursive nearest agglomeration rena unlike existing fast agglomerative schemes it avoids the creation of giant clusters we empirically validate that it approximates the data as well as traditional variance minimizing clustering schemes that have a quadratic complexity in addition we analyze signal approximation with feature clustering and show that it can remove noise improving subsequent analysis steps as a consequence data reduction by clustering features with rena yields very fast and accurate models enabling to process large datasets on budget our theoretical analysis is backed by extensive experiments on publicly available data that illustrate the computation efficiency and the denoising properties of the resulting dimension reduction scheme

correlation powered information engines and the thermodynamics of self correction information engines can use structured environments as a resource to generate work by randomizing ordered inputs and leveraging the increased shannon entropy to transfer energy from a thermal reservoir to a work reservoir we give a broadly applicable expression for the work production of an information engine generally modeled as a memoryful channel that communicates inputs to outputs as it interacts with an evolving environment the expression establishes that an information engine must have more than one memory state in order to leverage input environment correlations to emphasize this functioning we designed an information engine powered solely by temporal correlations and not by statistical biases as employed by previous engines key to this is the engine s ability to synchronize the engine automatically returns to a desired dynamical phase when thrown into an unwanted dissipative phase by corruptions in the input that is by unanticipated environmental fluctuations this self correcting mechanism is robust up to a critical level of corruption beyond which the system fails to act as an engine we give explicit analytical expressions for both work and critical corruption level and summarize engine performance via a thermodynamic function phase diagram over engine control parameters the results reveal a thermodynamic mechanism based on nonergodicity that underlies error correction as it operates to support resilient engineered and biological systems

computationally efficient reduced polynomial based algorithms for hermitian toeplitz matrices in this work the mathematical structure is analyzed that is associated with the fast order recursive algorithms for computing the reflection coefficients and the levinson polynomial associated with a hermitian positive definite toeplitz matrix a new form of three term recurrence relation is derived anda computationally efficient alternative to the classical levinson durbin algorithm is obtained the computational complexity of the new algorithm is the same as those of the split algorithms described in the recent literature the new algorithm also provides further insight into the mathematical properties of the structurally rich toeplitz matrices

clustering based on pairwise distances when the data is of mixed dimensions in the context of clustering we consider a generative model in a euclidean ambient space with clusters of different shapes dimensions sizes and densities in an asymptotic setting where the number of points becomes large we obtain theoretical guaranties for some emblematic methods based on pairwise distances a simple algorithm based on the extraction of connected components in a neighborhood graph hierarchical clustering with single linkage and the spectral clustering method of ng jordan and weiss the methods are shown to enjoy some near optimal properties in terms of separation between clusters and robustness to outliers the local scaling method of zelnik manor and perona is shown to lead to a near optimal choice for the scale in the first and third methods we also provide a lower bound on the spectral gap to consistently choose the correct number of clusters in the spectral method

split algorithms for skewsymmetric toeplitz matrices with arbitrary rank profile split levinson type and schur type algorithms for the solutions of linear systems with a nonsingular skewsymmetric toeplitz matrix are designed in contrast to previous ones the algorithms work for any nonsingular skewsymmetric toeplitz matrix moreover generalizations of zw and wz factorizations of skewsymmetric toeplitz matrices related to the new split algorithms are presented

jarzynski type equalities in gambling role of information in capital growth we study the capital growth in gambling with and without side information and memory effects we derive several equalities for gambling which are of similar form to the jarzynski equality and its extension to systems with feedback controls those relations provide us with new measures to quantify the effects of information on the statistics of capital growth in gambling we discuss the implications of the equalities and show that they reproduce the known upper bounds of average capital growth rates

thermodynamics of information processing based on enzyme kinetics an exactly solvable model of an information pump motivated by the recent proposed models of the information engine proc natl acad sci usa 109 11641 2012 and the information refrigerator phys rev lett 111 030602 2013 we propose a minimal model of the information pump and the information eraser based on enzyme kinetics this device can either pump molecules against the chemical potential gradient by consuming the information to be encoded in the bit stream or partially erase the information initially encoded in the bit stream by consuming the gibbs free energy the dynamics of this model is solved exactly and the phase diagram of the operation regimes is determined the efficiency and the power of the information machine is analyzed the validity of the second law of thermodynamics within our model is clarified our model offers a simple paradigm for the investigating of the thermodynamics of information processing involving the chemical potential in small systems

split algorithms for hermitian toeplitz matrices with arbitrary rank profile abstract split algorithms for toeplitz matrices exploit besides the toeplitz structure additional symmetry properties to reduce the number of operations in this paper split levinson and schur algorithms for hermitian toeplitz matrices are presented that work in contrast to previous algorithms without additional conditions like strong nonsingularity the main contribution is the generalization of the split levinson type algorithms of b krishna h krishna and h krishna s morgera to general nonsingular hermitian toeplitz matrices furthermore a schur type counterpart of this algorithm is presented that is also new in the strongly nonsingular case some auxiliary considerations concerning the kernel structure of hermitian toeplitz matrices might be of independent interest

dft representations of toeplitz plus hankel bezoutians with application to fast matrixvector multiplication abstract representations for inverses of toeplitz plus hankel matrices and more general t h bezoutians with discrete fourier transformations are presented which can be used for fast matrix vector multiplication with the help of some of the formulas multiplication by a toeplitz plus hankel matrix inverse can be carried out with six dft s plus eight dft s for preprocessing which improves the so far best known result of seven dft s plus 10 dft s for preprocessing in e bozzo linear algebra appl 230 1995 127 150

a two step even odd split levinson algorithm for toeplitz systems abstract we use a two step durbin method rather than the single step version in the even odd split levinson algorithm for strongly nonsingular real symmetric toeplitz systems with arbitrary right hand side thereby slightly reducing the complexity of this algorithm to 19 8 n 2 o n flops on a sequential machine we also present extensive numerical results comparing several levinson type methods

maximum work extraction and implementation costs for nonequilibrium maxwell s demons we determine the maximum amount of work extractable in finite time by a demon performing continuous measurements on a quadratic hamiltonian system subjected to thermal fluctuations in terms of the information extracted from the system the maximum work demon is found to apply a high gain continuous feedback involving a kalman bucy estimate of the system state and operates in nonequilibrium a simple and concrete electrical implementation of the feedback protocol is proposed which allows for analytic expressions of the flows of energy entropy and information inside the demon this let us show that any implementation of the demon must necessarily include an external power source which we prove both from classical thermodynamics arguments and from a version of landauer s memory erasure argument extended to nonequilibrium linear systems

optimal construction of k nearest neighbor graphs for identifying noisy clusters we study clustering algorithms based on neighborhood graphs on a random sample of data points the question we ask is how such a graph should be constructed in order to obtain optimal clustering results which type of neighborhood graph should one choose mutual k nearest neighbor or symmetric k nearest neighbor what is the optimal parameter k in our setting clusters are defined as connected components of the t level set of the underlying probability distribution clusters are said to be identified in the neighborhood graph if connected components in the graph correspond to the true underlying clusters using techniques from random geometric graph theory we prove bounds on the probability that clusters are identified successfully both in a noise free and in a noisy setting those bounds lead to several conclusions first k has to be chosen surprisingly high rather of the order n than of the order logn to maximize the probability of cluster identification secondly the major difference between the mutual and the symmetric k nearest neighbor graph occurs when one attempts to detect the most significant cluster only

fluctuation theorem with information exchange role of correlations in stochastic thermodynamics we establish the fluctuation theorem in the presence of information exchange between a nonequilibrium system and other degrees of freedom such as an observer and a feedback controller where the amount of information exchange is added to the entropy production the resulting generalized second law sets the fundamental limit of energy dissipation and energy cost during the information exchange our results apply not only to feedback controlled processes but also to a much broader class of information exchanges and provide a unified framework of nonequilibrium thermodynamics of measurement and feedback control

fluctuation response inequality out of equilibrium we present an approach to response around arbitrary out of equilibrium states in the form of a fluctuation response inequality fri we study the response of an observable to a perturbation of the underlying stochastic dynamics we find that the magnitude of the response is bounded from above by the fluctuations of the observable in the unperturbed system and the kullback leibler divergence between the probability densities describing the perturbed and the unperturbed system this establishes a connection between linear response and concepts of information theory we show that in many physical situations the relative entropy may be expressed in terms of physical observables as a direct consequence of this fri we show that for steady state particle transport the differential mobility is bounded by the diffusivity for a virtual perturbation proportional to the local mean velocity we recover the thermodynamic uncertainty relation tur for steady state transport processes finally we use the fri to derive a generalization of the uncertainty relation to arbitrary dynamics which involves higher order cumulants of the observable we provide an explicit example in which the tur is violated but its generalization is satisfied with equality

consistent procedures for cluster tree estimation and pruning for a density f on mathbb r d a high density cluster is any connected component of x f x geq lambda for some lambda 0 the set of all high density clusters forms a hierarchy called the cluster tree of f we present two procedures for estimating the cluster tree given samples from f the first is a robust variant of the single linkage algorithm for hierarchical clustering the second is based on the k nearest neighbor graph of the samples we give finite sample convergence rates for these algorithms which also imply consistency and we derive lower bounds on the sample complexity of cluster tree estimation finally we study a tree pruning procedure that guarantees under milder conditions than usual to remove clusters that are spurious while recovering those that are salient

generalized jarzynski equality under nonequilibrium feedback control the jarzynski equality is generalized to situations in which nonequilibrium systems are subject to a feedback control the new terms that arise as a consequence of the feedback describe the mutual information content obtained by measurement and the efficacy of the feedback control our results lead to a generalized fluctuation dissipation theorem that reflects the readout information and they can be experimentally tested using small thermodynamic systems we illustrate our general results by introducing an information ratchet which can transport a brownian particle in one direction and extract positive work from the particle

interpretations of directed information in portfolio theory data compression and hypothesis testing we investigate the role of directed information in portfolio theory data compression and statistics with causality constraints in particular we show that directed information is an upper bound on the increment in growth rates of optimal portfolios in a stock market due to causal side information this upper bound is tight for gambling in a horse race which is an extreme case of stock markets directed information also characterizes the value of causal side information in instantaneous compression and quantifies the benefit of causal inference in joint compression of two stochastic processes in hypothesis testing directed information evaluates the best error exponent for testing whether a random process y causally influences another process x or not these results lead to a natural interpretation of directed information i yn xn as the amount of information that a random sequence yn y1 y2 yn causally provides about another random sequence xn x1 x2 xn a new measure directed lautum information is also introduced and interpreted in portfolio theory data compression and hypothesis testing

consistency of single linkage for high density clusters abstract high density clusters are defined on a population with density f in r dimensions to be the maximal connected sets of form x f x c single linkage clustering is evaluated for consistency in detecting such high density clusters other standard hierarchical techniques such as average and complete linkage are hopelessly inconsistent for these clusters the asymptotic consistency of single linkage closely depends on the percolation problem of broadbent and hammersley if small spheres are removed at random from a solid at which density of spheres will water begin to flow through the solid if there is a single critical density such that no flow takes place below a certain density and flow occurs through a single connected set above that density then single linkage is consistent in separating high density clusters by disjoint single linkage clusters that include a positive fraction of sample points in the respective clusters and pass arbitrarily close to all points in the respective clusters

unifying three perspectives on information processing in stochastic thermodynamics so far feedback driven systems have been discussed using i measurement and control ii a tape interacting with a system or iii by identifying an implicit maxwell demon in steady state transport we derive the corresponding second laws from one master fluctuation theorem and discuss their relationship in particular we show that both the entropy production involving mutual information between system and controller and the one involving a shannon entropy difference of an information reservoir like a tape carry an extra term different from the usual current times affinity we thus generalize stochastic thermodynamics to the presence of an information reservoir

nonequilibrium fluctuation theorem for systems under discrete and continuous feedback control in the time reverse process of a feedback manipulated stochastic system we allow performing measurements without violating causality as a result we come across an entropy production due to the measurement process this entropy production in addition to the usual system and medium entropy production constitutes the total entropy production of the combined system of the reservoir the system and the feedback controller we show that this total entropy production of full system satisfies an integrated fluctuation theorem as well as a detailed fluctuation theorem as expected we illustrate and verify this idea through explicit calculation and direct simulation in two examples

many roads to synchrony natural time scales and their algorithms we consider two important time scales the markov and cryptic orders that monitor how an observer synchronizes to a finitary stochastic process we show how to compute these orders exactly and that they are most efficiently calculated from the e machine a process s minimal unifilar model surprisingly though the markov order is a basic concept from stochastic process theory it is not a probabilistic property of a process rather it is a topological property and moreover it is not computable from any finite state model other than the e machine via an exhaustive survey we close by demonstrating that infinite markov and infinite cryptic orders are a dominant feature in the space of finite memory processes we draw out the roles played in statistical mechanical spin systems by these two complementary length scales