In recent years there has been a growing interest in the study of sparse representation of signals. Using an overcomplete dictionary that contains prototype signal-atoms, signals are described by sparse linear combinations of these atoms. Applications that use sparse representation are many and include compression, regularization in inverse problems, feature extraction, and more. Recent activity in this field has concentrated mainly on the study of pursuit algorithms that decompose signals with respect to a given dictionary. Designing dictionaries to better fit the above model can be done by either selecting one from a prespecified set of linear transforms or adapting the dictionary to a set of training signals. Both of these techniques have been considered, but this topic is largely still open. In this paper we propose a novel algorithm for adapting dictionaries in order to achieve sparse signal representations. Given a set of training signals, we seek the dictionary that leads to the best representation for each member in this set, under strict sparsity constraints. We present a new method-the K-SVD algorithm-generalizing the K-means clustering process. K-SVD is an iterative method that alternates between sparse coding of the examples based on the current dictionary and a process of updating the dictionary atoms to better fit the data. The update of the dictionary columns is combined with an update of the sparse representations, thereby accelerating convergence. The K-SVD algorithm is flexible and can work with any pursuit method (e.g., basis pursuit, FOCUSS, or matching pursuit). We analyze this algorithm and demonstrate its results both on synthetic tests and in applications on real image data

https://users.ece.utexas.edu/~sanghavi/courses/papers/KSVD.pdf

$rm K$ -SVD: An Algorithm for Designing Overcomplete Dictionaries for Sparse Representation

This paper demonstrates theoretically and empirically that a greedy algorithm called orthogonal matching pursuit (OMP) can reliably recover a signal with m nonzero entries in dimension d given O(m ln d) random linear measurements of that signal. This is a massive improvement over previous results, which require O(m2) measurements. The new results for OMP are comparable with recent results for another approach called basis pursuit (BP). In some settings, the OMP algorithm is faster and easier to implement, so it is an attractive alternative to BP for signal recovery problems.

/pdf/signal-recovery-from-random-measurements-via-orthogonal-5adhd8dwmq.pdf

Signal Recovery From Random Measurements Via Orthogonal Matching Pursuit

This report demonstrates theoretically and empirically that a greedy algorithm called
Orthogonal Matching Pursuit (OMP) can reliably recover a signal with m nonzero entries in dimension
d given O(mln d) random linear measurements of that signal. This is a massive improvement
over previous results, which require O(m2) measurements. The new results for OMP are comparable
with recent results for another approach called Basis Pursuit (BP). In some settings, the
OMP algorithm is faster and easier to implement, so it is an attractive alternative to BP for signal
recovery problems.

/pdf/signal-recovery-from-random-measurements-via-orthogonal-2uks7d8txo.pdf

Signal Recovery from Random Measurements Via Orthogonal Matching Pursuit: The Gaussian Case

Optimal shape design of structural elements based on boundary variations results in final designs that are topologically equivalent to the initial choice of design, and general, stable computational schemes for this approach often require some kind of remeshing of the finite element approximation of the analysis problem. This paper presents a methodology for optimal shape design where both these drawbacks can be avoided. The method is related to modern production techniques and consists of computing the optimal distribution in space of an anisotropic material that is constructed by introducing an infimum of periodically distributed small holes in a given homogeneous, i~otropic material, with the requirement that the resulting structure can carry the given loads as well as satisfy other design requirements. The computation of effective material properties for the anisotropic material is carried out using the method of homogenization. Computational results are presented and compared with results obtained by boundary variations.

/pdf/generating-optimal-topologies-in-structural-design-using-a-2ul70wm6io.pdf

Generating optimal topologies in structural design using a homogenization method

Convergence of Probability Measures. By P. Billingsley. Chichester, Sussex, Wiley, 1968. xii, 253 p. 9 1/4“. 117s.

Convergence of Probability Measures

The problem of optimally approximating a function with a linear expansion over a redundant dictionary of waveforms is NP-hard. The greedy matching pursuit algorithm and its orthogonalized variant produce suboptimal function expansions by iteratively choosing dictionary waveforms that best match the function’s structures. A matching pursuit provides a means of quickly computing compact, adaptive function approximations. Numerical experiments show that the approximation errors from matching pursuits initially decrease rapidly, but the asymptotic decay rate of the errors is slow. We explain this behavior by showing that matching pursuits are chaotic, ergodic maps. The statistical properties of the approximation errors of a pursuit can be obtained from the invariant measure of the pursuit. We characterize these measures using group symmetries of dictionaries and by constructing a stochastic differential equation model. We derive a notion of the coherence of a signal with respect to a dictionary from our characterization of the approximation errors of a pursuit. The dictionary elements slected during the initial iterations of a pursuit correspond to a function’s coherent structures. The tail of the expansion, on the other hand, corresponds to a noise which is characterized by the invariant measure of the pursuit map. When using a suitable dictionary, the expansion of a function into its coherent structures yields a compact approximation. We demonstrate a denoising algorithm based on coherent function expansions.

Adaptive greedy approximations

We present a model for pricing and hedging derivative securities and option portfolios in an environment where the volatility is not known precisely, but is assumed instead to lie between two extreme values σminand σmax. These bounds could be inferred from extreme values of the implied volatilities of liquid options, or from high-low peaks in historical stock- or option-implied volatilities. They can be viewed as defining a confidence interval for future volatility values. We show that the extremal non-arbitrageable prices for the derivative asset which arise as the volatility paths vary in such a band can be described by a non-linear PDE, which we call the Black-Scholes-Barenblatt equation. In this equation, the ‘pricing’ volatility is selected dynamically from the two extreme values, σmin, σmax, according to the convexity of the value-function. A simple algorithm for solving the equation by finite-differencing or a trinomial tree is presented. We show that this model captures the importance of diversifi...

http://janroman.dhis.org/finance/Hedgeing/Priceing%20and%20Hedgeing%20with%20unknown%20Volatilities.pdf

Pricing and hedging derivative securities in markets with uncertain volatilities

The role of a dealer in securities markets is to provide liquidity on the exchange by quoting bid and ask prices at which he is willing to buy and sell a specific quantity of assets. Traditionally,...

/pdf/high-frequency-trading-in-a-limit-order-book-1ckbzw5782.pdf

High-frequency trading in a limit order book

Compactness methods in the theory of homogenization

We consider the magnetoelectric effect arising in a multilayer composite consisting of bonded layers of a piezoelectric ceramic and a magnetostrictive material operating in the linear regime. Magnetoelectric coupling arises from the mechanical contacts between layers and the elec tric/mechanical and magnetic/mechanical coupling in each phase. Calculations of the magnetoelec tric coefficient α* as well as of the figures of merit kme and kme,cl are presented. These calculations fully take into account the electric, magnetic and mechanical mismatch in the material properties, as well as the volume-fractions of both materials. We derive the optimal volume fraction fcrit1 of piezoelectric material needed to maximize the figure of merit k fme or kme,cl for the composite and give criteria for optimizing the magnetoelectric effect in terms of the choice of the individual constit uents. We computed the figures of merit of a variety of piezoelectric/magnetostrictive combinations. The combinations CoFe2O4/PZT5H (kme...

Marco Avellaneda

Papers

Adaptive greedy approximations

Pricing and hedging derivative securities in markets with uncertain volatilities

High-frequency trading in a limit order book

Compactness methods in the theory of homogenization

Magnetoelectric Effect in Piezoelectric/Magnetostrictive Multilayer (2-2) Composites