MATHICSE Technical Report : Accelerated filtering on graphs using Lanczos method

Sinal-processing on graphs has developed into a very active field of research during the last decade. In particular, the number of applications using frames con-structed from graphs, like wavelets on graphs, has substantially increased. To attain scalability for large graphs, fast graph-signal filtering techniques are needed. In this contribution, we propose an accelerated algorithm based on the Lanczos method that adapts to the Laplacian spectrum without explicitly computing it. The result is an accurate, robust, scalable and efficient algorithm. Compared to existing methods based on Chebyshev polynomials, our solution achieves higher accuracy without increasing the overall complexity significantly. Furthermore, it is particularly well suited for graphs with large spectral gaps.

Numerical methods for option pricing: polynomial approximation and high dimensionality

Francesco Statti

Options are some of the most traded financial instruments and computing their price is a central task in financial mathematics and in practice. Consequently, the development of numerical algorithms for pricing options is an active field of research. In general, evaluating the price of a specific option relies on the properties of the stochastic model used for the underlying asset price. In this thesis we develop efficient and accurate numerical methods for option pricing in a specific class of models: polynomial models. They are a versatile tool for financial modeling and have useful properties that can be exploited for option pricing. Significant challenges arise when developing option pricing techniques. For instance, the underlying model might have a high-dimensional parameter space. Furthermore, treating multi-asset options yields high-dimensional pricing problems. Therefore, the pricing method should be able to handle high dimensionality. Another important aspect is the efficiency of the algorithm: in real-world applications, option prices need to be delivered within short periods of time, making the algorithmic complexity a potential bottleneck. In this thesis, we address these challenges by developing option pricing techniques that are able to handle low and high-dimensional problems, and we propose complexity reduction techniques. The thesis consists of four parts: First, we present a methodology for European and American option pricing. The method uses the moments of the underlying price process to produce monotone sequences of lower and upper bounds of the option price. The bounds are obtained by solving a sequence of polynomial optimization problems. As the order of the moments increases, the bounds become sharper and eventually converge to the exact price under appropriate assumptions. Second, we develop a fast algorithm for the incremental computation of nested block triangular matrix exponentials. This algorithm allows for an efficient incremental computation of the moment sequence of polynomial jump-diffusions. In other words, moments of order 0, 1, 2, 3... are computed sequentially until a dynamically evaluated criterion tells us to stop. The algorithm is based on the scaling and squaring technique and reduces the complexity of the pricing algorithms that require such an incremental moment computation. Third, we develop a complexity reduction technique for high-dimensional option pricing. To this end, we first consider the option price as a function of model and payoff parameters. Then, the tensorized Chebyshev interpolation is used on the parameter space to increase the efficiency in computing option prices, while maintaining the required accuracy. The high dimensionality of the problem is treated by expressing the tensorized interpolation in the tensor train format and by deriving an efficient way, which is based on tensor completion, to approximate the interpolation coefficients. Lastly, we propose a methodology for pricing single and multi-asset European options. The approach is a combination of Monte Carlo simulation and function approximation. We address the memory limitations that arise when treating very high-dimensional applications by combining the method with optimal sampling strategies and using a randomized algorithm to reduce the storage complexity of the approach. The obtained numerical results show the effectiveness of the algorithms developed in this thesis.

EPFL2019

An Adaptive Sublinear-Time Block Sparse Fourier Transform

Volkan Cevher, Mikhail Kapralov, Jonathan Mark Scarlett, Amir Zandieh

The problem of approximately computing the

k

dominant Fourier coefficients of a vector

X

quickly, and using few samples in time domain, is known as the Sparse Fourier Transform (sparse FFT) problem. A long line of work on the sparse FFT has resulted in algorithms with

O(k\log n\log (n/k))

runtime [Hassanieh \emph{et al.}, STOC'12] and

O(k\log n)

sample complexity [Indyk \emph{et al.}, FOCS'14]. These results are proved using non-adaptive algorithms, and the latter

O(k\log n)

sample complexity result is essentially the best possible under the sparsity assumption alone: It is known that even adaptive algorithms must use

\Omega((k\log(n/k))/\log\log n)

samples [Hassanieh \emph{et al.}, STOC'12]. By {\em adaptive}, we mean being able to exploit previous samples in guiding the selection of further samples. This paper revisits the sparse FFT problem with the added twist that the sparse coefficients approximately obey a

(k_0,k_1)

-block sparse model. In this model, signal frequencies are clustered in

k_0

intervals with width

k_1

in Fourier space, and

k= k_0k_1

is the total sparsity. Signals arising in applications are often well approximated by this model with

k_0\ll k

. Our main result is the first sparse FFT algorithm for

(k_0, k_1)

-block sparse signals with a sample complexity of

O^*(k_0k_1 + k_0\log(1+ k_0)\log n)

at constant signal-to-noise ratios, and sublinear runtime. A similar sample complexity was previously achieved in the works on {\em model-based compressive sensing} using random Gaussian measurements, but used

\Omega(n)

runtime. To the best of our knowledge, our result is the first sublinear-time algorithm for model based compressed sensing, and the first sparse FFT result that goes below the

O(k\log n)

sample complexity bound. Interestingly, the aforementioned model-based compressive sensing result that relies on Gaussian measurements is non-adaptive, whereas our algorithm crucially uses {\em adaptivity} to achieve the improved sample complexity bound. We prove that adaptivity is in fact necessary in the Fourier setting: Any {\em non-adaptive} algorithm must use

\Omega(k_0k_1\log \frac{n}{k_0k_1})

samples for the

(k_0,k_1

)-block sparse model, ruling out improvements over the vanilla sparsity assumption. Our main technical innovation for adaptivity is a new randomized energy-based importance sampling technique that may be of independent interest.

2017