optimum sequence multisets for synchronous code division multiple access channels it is shown that the sum capacity of the symbol synchronous code division multiple access channel with equal average input energy constraints is maximized precisely by those spreading sequence multisets that meet welch s lower bound on total squared correlation it is further shown that the symmetric capacity of the channel determined by these same sequence multisets is equal to the sum capacity

time analyticity for solutions of nonlinear abstract parabolic evolution equations we study analyticity with respect to time of the solutions to a class of nonlinear evolution equations in banach space the results are applied to cauchy dirichlet problems for fully nonlinear parabolic integrodifferential equations such as ut t x f t x u t x u t x u t x 0t g t s x u s x u s x u s c ds t 0 x gw where is a regular bounded open set in rn

optimal phase transitions in compressed sensing compressed sensing deals with efficient recovery of analog signals from linear encodings this paper presents a statistical study of compressed sensing by modeling the input signal as an i i d process with known distribution three classes of encoders are considered namely optimal nonlinear optimal linear and random linear encoders focusing on optimal decoders we investigate the fundamental tradeoff between measurement rate and reconstruction fidelity gauged by error probability and noise sensitivity in the absence and presence of measurement noise respectively the optimal phase transition threshold is determined as a functional of the input distribution and compared to suboptimal thresholds achieved by popular reconstruction algorithms in particular we show that gaussian sensing matrices incur no penalty on the phase transition threshold with respect to optimal nonlinear encoding our results also provide a rigorous justification of previous results based on replica heuristics in the weak noise regime

signals on graphs uncertainty principle and sampling in many applications the observations can be represented as a signal defined over the vertices of a graph the analysis of such signals requires the extension of standard signal processing tools in this paper first we provide a class of graph signals that are maximally concentrated on the graph domain and on its dual then building on this framework we derive an uncertainty principle for graph signals and illustrate the conditions for the recovery of band limited signals from a subset of samples we show an interesting link between uncertainty principle and sampling and propose alternative signal recovery algorithms including a generalization to frame based reconstruction methods after showing that the performance of signal recovery algorithms is significantly affected by the location of samples we suggest and compare a few alternative sampling strategies finally we provide the conditions for perfect recovery of a useful signal corrupted by sparse noise showing that this problem is also intrinsically related to vertex frequency localization properties

time analyticity of solutions to the ricci flow we prove that if g t is a smooth complete solution to the ricci flow of uniformly bounded curvature on m times 0 omega then the correspondence t mapsto g t is real analytic at each t 0 in 0 omega the analyticity is a consequence of classical bernstein type estimates on the temporal and spatial derivatives of the curvature tensor which we further use to show that under the above global hypotheses for any x 0 in m and t 0 in 0 omega there exist local coordinates x x i on a neighborhood u subset m of x 0 in which the representation g ij x t of the metric is real analytic in both x and t on some cylinder u times t 0 epsilon t 0 epsilon

convergence of double fourier series and w classes the double fourier series of functions of the generalized bounded variation class n ln n 1 bv are shown to be pringsheim convergent everywhere in a certain sense this result cannot be improved in general functions of class a bv defined here have quadrant limits at every point and for f e a bv there exist at most countable sets p and q such that for x p and y q f is continuous at x y it is shown that the previously studied class abv contains essentially discontinuous functions unless the sequence a satisfies a strong condition

near optimal sensor placement for linear inverse problems a classic problem is the estimation of a set of parameters from measurements collected by only a few sensors the number of sensors is often limited by physical or economical constraints and their placement is of fundamental importance to obtain accurate estimates unfortunately the selection of the optimal sensor locations is intrinsically combinatorial and the available approximation algorithms are not guaranteed to generate good solutions in all cases of interest we propose framesense a greedy algorithm for the selection of optimal sensor locations the core cost function of the algorithm is the frame potential a scalar property of matrices that measures the orthogonality of its rows notably framesense is the first algorithm that is near optimal in terms of mean square error meaning that its solution is always guaranteed to be close to the optimal one moreover we show with an extensive set of numerical experiments that framesense achieves state of the art performance while having the lowest computational cost when compared to other greedy methods

information based objective functions for active data selection learning can be made more efficient if we can actively select particularly salient data points within a bayesian learning framework objective functions are discussed that measure the expected informativeness of candidate measurements three alternative specifications of what we want to gain information about lead to three different criteria for data selection all these criteria depend on the assumption that the hypothesis space is correct which may prove to be their main weakness

a characterization of quality adjusted life years under cumulative prospect theory quality adjusted life years qalys are the most common utility measure in medical decision analysis and economic evaluations of health care this paper presents an axiomatization of qalys under cumulative prospect theory cpt currently the most influential model for decision under uncertainty because the set of health states need not be endowed with a natural topology that is connected we first show how existing cpt characterizations can be extended to a class of outcome sets for which no connected natural topology is given we then characterize qaly models with linear power and exponential utility for duration finally we define loss aversion for multiattribute utility theory and characterize the qaly models under general and constant loss aversion the measurement of qalys belongs to the general field of multiattribute utility theory hence our results can be generalized to other multiattribute decision contexts and they thereby contribute to the development of multiattribute utility theory under cumulative prospect theory

modeling uncertainty in multi criteria decision analysis abstract his paper provides a review of multiple criteria decision analysis mcda for cases where attribute evaluations are uncertain the main aim is to identify different tools which can be used to represent uncertain evaluations and to broadly survey the available decision models that can be used to support uncertain decision making the review includes models using probabilities or probability like quantities explicit risk measures such as quantiles and variances fuzzy numbers and scenarios the practical assessment of uncertain outcomes and preferences associated with these outcomes is also discussed

cumulative prospect theory for parametric and multiattribute utilities different attitudes towards gains and losses are a prominent feature of cumulative prospect theory for decision under uncertainty in particular decision weights for uncertain events can depend on whether the events involve gains or losses and the shape of the utility function can reveal loss aversion decision analyses concentrate on event capacities which determine decision weights and on the shape of the utility function the present paper focuses on linear exponential power function and multilinear utility models for decision under uncertainty we begin with straightforward preference axioms for a representation by a cumulative prospect theory functional the axioms include weak ordering continuity monotonicity and tail independence we show that in their presence constant absolute proportional risk aversion implies linear exponential power utility then for the multiattribute case mutual utility independence leads to a utility function that is additive multiplicative multilinear

distortion rate function of sub nyquist sampled gaussian sources the amount of information lost in sub nyquist sampling of a continuous time gaussian stationary process is quantified we consider a combined source coding and sub nyquist reconstruction problem in which the input to the encoder is a noisy sub nyquist sampled version of the analog source we first derive an expression for the mean squared error in the reconstruction of the process from a noisy and information rate limited version of its samples this expression is a function of the sampling frequency and the average number of bits describing each sample it is given as the sum of two terms minimum mean square error in estimating the source from its noisy but otherwise fully observed sub nyquist samples and a second term obtained by reverse waterfilling over an average of spectral densities associated with the polyphase components of the source we extend this result to multi branch uniform sampling where the samples are available through a set of parallel channels with a uniform sampler and a pre sampling filter in each branch further optimization to reduce distortion is then performed over the pre sampling filters and an optimal set of pre sampling filters associated with the statistics of the input signal and the sampling frequency is found this results in an expression for the minimal possible distortion achievable under any analog to digital conversion scheme involving uniform sampling and linear filtering these results thus unify the shannon whittaker kotelnikov sampling theorem and shannon rate distortion theory for gaussian sources

reconstructing gaussian sources by spatial sampling consider a gaussian memoryless multiple source gmms with m components with joint probability distribution known only to lie in a given class of distributions a subset of k leq m components is sampled and compressed with the objective of reconstructing all the m components within a specified level of distortion under a mean squared error criterion in bayesian and nonbayesian settings the notion of universal sampling rate distortion function for gaussian sources is introduced to capture the optimal tradeoffs among sampling compression rate and distortion level single letter characterizations are provided for the universal sampling rate distortion function our achievability proofs highlight the following structural property it is optimal to compress and reconstruct first the sampled components of the gmms alone and then form estimates for the unsampled components based on the former

sensor placement by maximal projection on minimum eigenspace for linear inverse problems this paper presents two new greedy sensor placement algorithms named minimum nonzero eigenvalue pursuit mnep and maximal projection on minimum eigenspace mpme for linear inverse problems with greater emphasis on the mpme algorithm for performance comparison with existing approaches in both mnep and mpme we select the sensing locations one by one in this way the least number of required sensor nodes can be determined by checking whether the estimation accuracy is satisfied after each sensing location is determined for the mpme algorithm the minimum eigenspace is defined as the eigenspace associated with the minimum eigenvalue of the dual observation matrix for each sensing location the projection of its observation vector onto the minimum eigenspace is shown to be monotonically decreasing w r t the worst case error variance wcev of the estimated parameters we select the sensing location whose observation vector has the maximum projection onto the minimum eigenspace of the current dual observation matrix the proposed mpme is shown to be one of the most computationally efficient algorithms our monte carlo simulations showed that mpme outperforms the convex relaxation method the sparsense method and the framesense method in terms of wcev and the mean square error mse of the estimated parameters especially when the number of available sensor nodes is very limited

subjective probability and expected utility without additivity comonotic acts are defined and an axiom of comonotomic independence 19 introduced that weakens the von newmann morgenstern axiom of independence if a nondegenerate continuous and monotonic state independent weak order over acts satisfies comonotonic independence then it induces a unique non necessarily additive probability and a von neumann morgenstern utility an extension of theexpected utility theory covers situations as the ellsberg paradox which are inconsistent with additive expected utility the concept of uncertainty aversion and interpretation of comonotonic independence in the context of social welfare functions are included

risk aversion in cumulative prospect theory this paper characterizes the conditions for strong risk aversion and second order stochastic dominance for cumulative prospect theory strong risk aversion implies a convex weighting function for gains and a concave one for losses it does not necessarily imply a concave utility function the latter does follow if the weighting functions are continuous by investigating the exact relationship between loss aversion and strong risk aversion a natural index for the degree of loss aversion is derived

sampling rate distortion consider a discrete memoryless multiple source with m components of which k leq m possibly different sources are sampled at each time instant and jointly compressed in order to reconstruct all the m sources under a given distortion criterion a new notion of sampling rate distortion function is introduced and is characterized first for the case of fixed set sampling next for independent random sampling performed without knowledge of the source outputs it is shown that the sampling rate distortion function is the same regardless of whether or not the decoder is informed of the sequence of sampled sets furthermore memoryless random sampling is considered with the sampler depending on the source outputs and with an informed decoder it is shown that deterministic sampling characterized by a conditional point mass is optimal and suffices to achieve the sampling rate distortion function for memoryless random sampling with an uninformed decoder an upper bound for the sampling rate distortion function is seen to possess a similar property of conditional point mass optimality it is shown by example that memoryless sampling with an informed decoder can outperform strictly any independent random sampler and that memoryless sampling can do strictly better with an informed decoder than without

generalized choquet like aggregation functions for handling bipolar scales we are interested in modeling interaction between criteria in multi criteria decision making when underlying scales are bipolar interacting phenomena involving behavioral bias between attractive and repulsive values are in particular considered here we show in an example that both the choquet integral and the cumulative prospect theory cpt model fail to represent these interacting phenomena axioms that enable the construction of the preferences of the decision maker over each attribute and the representation of his preferences about aggregation of criteria are introduced and justified we show there is a unique aggregation operator that fits with these axioms it is based on the notion of em bi capacity and generalizes both the choquet integral and the cpt model

analysis and design of optimum sparse array configurations for adaptive beamforming in this paper we analyze the effect of nonuniform array configurations on adaptive beamforming for enhanced signal to interference plus noise ratio sinr the array is configured using a given number of antennas or through a selection of subset of antennas from a larger available set leading to a sparse array in both cases the bounds on the highest achievable sinr for a given number of antennas are formulated and used to offer new insights into open loop adaptive beamforming the upper and lower bounds underscore the role of array configurations in optimizing performance for interference free and interference active environments respectively this paper considers the general case of multiple sources and interferences in the field of view we formulate three angles namely eigenspace angle conventional angle and minimum canonical angle for characterizing spatial separation between the source and interference subspaces which represent performance loss incurred by interference nulling three sparse array design methods incorporating these angles are proposed simulation examples confirm the role of subspace angles in optimum beamforming and validate the utility of antenna selection algorithms for sparse array design

fundamental distortion limits of analog to digital compression representing a continuous time signal by a set of samples is a classical problem in signal processing we study this problem under the additional constraint that the samples are quantized or compressed in a lossy manner under a limited bitrate budget to this end we consider a combined sampling and source coding problem in which an analog stationary gaussian signal is reconstructed from its encoded samples these samples are obtained by a set of bounded linear functionals of the continuous time path with a limitation on the average number of samples per unit time given in this setting we provide a full characterization of the minimal distortion in terms of the sampling frequency the bitrate and the signal s spectrum assuming that the signal s energy is not uniformly distributed over its spectral support we show that for each compression bitrate there exists a critical sampling frequency smaller than the nyquist rate such that the distortion in signal reconstruction when sampling at this frequency is minimal our results can be seen as an extension of the classical sampling theorem for bandlimited random processes in the sense that they describe the minimal amount of excess distortion in the reconstruction due to lossy compression of the samples and provide the minimal sampling frequency required in order to achieve this distortion finally we compare the fundamental limits in the combined source coding and sampling problem to the performance of pulse code modulation where each sample is quantized by a scalar quantizer using a fixed number of bits

sampling signals with finite rate of innovation the authors consider classes of signals that have a finite number of degrees of freedom per unit of time and call this number the rate of innovation examples of signals with a finite rate of innovation include streams of diracs e g the poisson process nonuniform splines and piecewise polynomials even though these signals are not bandlimited we show that they can be sampled uniformly at or above the rate of innovation using an appropriate kernel and then be perfectly reconstructed thus we prove sampling theorems for classes of signals and kernels that generalize the classic bandlimited and sinc kernel case in particular we show how to sample and reconstruct periodic and finite length streams of diracs nonuniform splines and piecewise polynomials using sinc and gaussian kernels for infinite length signals with finite local rate of innovation we show local sampling and reconstruction based on spline kernels the key in all constructions is to identify the innovative part of a signal e g time instants and weights of diracs using an annihilating or locator filter a device well known in spectral analysis and error correction coding this leads to standard computational procedures for solving the sampling problem which we show through experimental results applications of these new sampling results can be found in signal processing communications systems and biological systems

on sampling and coding for distributed acoustic sensing the issue of how to efficiently represent the data collected by a network of microphones recording spatio temporal acoustic wave fields is addressed each sensor node in the network samples the sound field quantizes the samples and transmits the encoded samples to some central unit which computes an estimate of the original sound field based on the information received from all the microphones our analysis is based on the spectral properties of the sound field which are induced by the physics of wave propagation and have a significant impact on the efficiency of the chosen sampling lattice and coding scheme as field acquisition by a sensor network typically implies spatio temporal sampling of the field a multidimensional sampling theorem for homogeneous random fields with compactly supported spectral measures is proved to assess the loss of information implied by source coding rate distortion functions for various coding schemes and sampling lattices are determined in particular centralized coding independent coding and some multiterminal schemes are compared under the assumption of spectral whiteness of the sound field it is shown that sampling with a quincunx lattice followed by independent coding is optimal as it achieves the lower bound given by centralized coding

on the abstract cauchy problem of parabolic type in spaces of continuous functions on etudie l existence l unicite et les proprietes de regularite des solutions du probleme de cauchy u t au r f t o t t u o u 0 ou a d a ce e est un operateur lineaire clos dans l espace de banach e

near optimal sensor placements in gaussian processes theory efficient algorithms and empirical studies when monitoring spatial phenomena which can often be modeled as gaussian processes gps choosing sensor locations is a fundamental task there are several common strategies to address this task for example geometry or disk models placing sensors at the points of highest entropy variance in the gp model and a d or e optimal design in this paper we tackle the combinatorial optimization problem of maximizing the mutual information between the chosen locations and the locations which are not selected we prove that the problem of finding the configuration that maximizes mutual information is np complete to address this issue we describe a polynomial time approximation that is within 1 1 e of the optimum by exploiting the submodularity of mutual information we also show how submodularity can be used to obtain online bounds and design branch and bound search procedures we then extend our algorithm to exploit lazy evaluations and local structure in the gp yielding significant speedups we also extend our approach to find placements which are robust against node failures and uncertainties in the model these extensions are again associated with rigorous theoretical approximation guarantees exploiting the submodularity of the objective function we demonstrate the advantages of our approach towards optimizing mutual information in a very extensive empirical study on two real world data sets

constant exchange risk properties this paper develops a methodology using risk properties to characterize the functional form of a utility measure for decision making under uncertainty the constant absolute risk property for example is known to be necessary and sufficient with appropriate regularity conditions for the utility function to have either a linear or an exponential form a new generalization of this property called the constant exchange risk property gives a characterization of six utility functions the linear function the exponential function the quadratic function the sum of two exponential functions the sum of a linear and an exponential function and the product of a linear and an exponential function since all of these functional forms have been used previously as approximations this methodology allows analysts to distinguish beforehand between alternative forms and thus properly specify the utility function in applications

analyticity of the maximal solution of an abstract nonlinear parabolic equation where u t x 0 7 j x r r satisfies other suitable conditions problem p is studied under the assumption that af au generates an analytic semigroup in e and has domain f which corresponds to a parabolicity condition on equation p problems of this kind have been studied by monotonicity methods see brezis 111 crandall ligget 3 and crandall pazy 4 to obtain global but weak solutions of p here we find a maximally defined strict solution by means of a linearization method and by virtue of maximal regularity results for the linear case this method has been used by da prato grisvard in 5 to obtain strict solutions of p with arguments which seem to be more simple and suitable to our purpose we get also the analyticity of u with respect to 1 w when f is analytic in section 1 some notation and assumptions are given problem p is studied in section 2 where we prove existence and uniqueness of the strict solution of p which is defined on a maximal time interval contained in 0 t and it is continuous with respect to t 0 in section 3 we give conditions to get global existence of the solution in section 4 we study analyticity with respect to t ug of the maximally defined solution of p and in section 5 a simple example is given in a subsequent paper we will show how this theory applies to the study of the analyttctty of the solutions of nonlinear parabol ic partial differential equations in spaces of holder continuous functions

rate distortion behavior of sparse sources the rate distortion behavior of sparse memoryless sources is studied these serve as models of sparse signal representations and facilitate the performance analysis of sparsifying transforms like the wavelet transform and nonlinear approximation schemes for strictly sparse binary sources with hamming distortion r d is shown to be almost linear for nonstrictly sparse continuous valued sources termed compressible two measures of compressibility are introduced incomplete moments and geometric mean the former lead to low and high rate upper bounds on mean squared error d r while the latter yields lower and upper bounds on source entropy thereby characterizing asymptotic r d behavior thus the notion of compressibility is quantitatively connected with actual lossy compression these bounding techniques are applied to two source models gaussian mixtures and power laws matching the approximately scale invariant decay of wavelet coefficients the former are versatile models for sparse data which in particular allow to bound high rate compression performance of a scalar mixture compared to a corresponding unmixed transform coding system such a comparison is interesting for transforms with known coefficient decay but unknown coefficient ordering e g when positions of highest variance coefficients are unknown the use of these models and results in distributed coding and compressed sensing scenarios are also discussed

sampling high dimensional bandlimited fields on low dimensional manifolds consider the task of sampling and reconstructing a bandlimited spatial field in r2 using moving sensors that take measurements along their path it is inexpensive to increase the sampling rate along the paths of the sensors but more expensive to increase the total distance traveled by the sensors per unit area which we call the path density in this paper we introduce the problem of designing sensor trajectories that are minimal in path density subject to the condition that the measurements of the field on these trajectories admit perfect reconstruction of bandlimited fields we study various possible designs of sampling trajectories generalizing some ideas from the classical theory of sampling on lattices we obtain necessary and sufficient conditions on the trajectories for perfect reconstruction we show that a single set of equispaced parallel lines has the lowest path density from certain restricted classes of trajectories that admit perfect reconstruction we then generalize some of our results to higher dimensions we first obtain results on designing sampling trajectories in higher dimensional fields further interpreting trajectories as 1 d manifolds we extend some of our ideas to higher dimensional sampling manifolds we formulate the problem of designing dimensional sampling manifolds for d dimensional spatial fields that are minimal in manifold density a natural generalization of the path density we show that our results on sampling trajectories for fields in r2 can be generalized to analogous results on d dimensional sampling manifolds for d dimensional spatial fields

fully nonlinear integrodifferential equations in general banach space on etudie des equations integrodifferentielles non lineaires dans un espace de banach qui peuvent etre considerees comme la version abstraite d equations integrodifferentielles partielles non lineaires on considere un probleme aux valeurs limites et initiales

the rate distortion dimension of sets and measures data compression of independent samples drawn from a fractal set is considered the asymptotic ratio of rate to magnitude log distortion characterizes the effective dimension occupied by the underlying distribution this quantity is shown to be identical to renyi s 1959 information dimension for self similar fractal sets this dimension is distribution dependent in sharp contrast with the behavior of absolutely continuous measures the rate distortion dimension of a set is defined as the maximal rate distortion dimension for distributions supported on this set kolmogorov s metric dimension is an upper bound on the rate distortion dimension while the hausdorff dimension is a lower bound examples of sets for which the rate distortion dimension differs from these bounds are provided

the distortion rate function of sampled wiener processes we consider the recovery of a continuous time wiener process from a quantized or a lossy compressed version of its uniform samples under limited bitrate and sampling rate we derive a closed form expression for the optimal tradeoff among sampling rate bitrate and quadratic distortion in this setting this expression is given in terms of a reverse waterfilling formula over the asymptotic spectral distribution of a sequence of finite rank operators associated with the optimal estimator of the wiener process from its samples we show that the ratio between this expression and the standard distortion rate function of the wiener process describing the optimal tradeoff between bitrate and distortion without a sampling constraint is only a function of the number of bits per sample we also consider a sub optimal lossy compression scheme in which the continuous time process is estimated from the output of an encoder that is optimal with respect to the discrete time samples we show that the latter is strictly greater than the distortion under optimal encoding but only by at most 3 we therefore conclude that near optimal performance is attained even if the encoder is unaware of the continuous time origin of the samples