Lifts of convex sets and cone factorizations

In this paper we address the basic geometric question of when a given convex set is the image under a linear map of an affine slice of a given closed convex cone. Such a representation or 'lift' of the convex set is especially useful if the cone admits an efficient algorithm for linear optimization over its affine slices. We show that the existence of a lift of a convex set to a cone is equivalent to the existence of a factorization of an operator associated to the set and its polar via elements in the cone and its dual. This generalizes a theorem of Yannakakis that established a connection between polyhedral lifts of a polytope and nonnegative factorizations of its slack matrix. Symmetric lifts of convex sets can also be characterized similarly. When the cones live in a family, our results lead to the definition of the rank of a convex set with respect to this family. We present results about this rank in the context of cones of positive semidefinite matrices. Our methods provide new tools for understanding cone lifts of convex sets.Comment: 20 pages, 2 figure

On the maximal order of numbers in the “factorisatio numerorum” problem

AbstractLet m(n) be the number of ordered factorizations of n⩾1 in factors larger than 1. We prove that for every ε>0m(n)<nρexp((logn)1/ρ/(loglogn)1+ε) holds for all integers n>n0, while, for a suitable constant c>0,m(n)>nρexp(c(logn/loglogn)1/ρ) holds for infinitely many positive integers n, where ρ=1.72864… is the positive real solution to ζ(ρ)=2. We investigate also arithmetic properties of m(n) and the number of distinct values of m(n)

OPERATOR METHODS, ABELIAN PROCESSES AND DYNAMIC CONDITIONING

A mathematical framework for Continuous Time Finance based on operator algebraic methods oers a new direct and entirely constructive perspective on the field. It also leads to new numerical analysis techniques which can take advantage of the emerging massively parallel GPU architectures which are uniquely suited to execute large matrix manipulations. This is partly a review paper as it covers and expands on the mathematical framework underlying a series of more applied articles. In addition, this article also presents a few key new theorems that make the treatment self-contained. Stochastic processes with continuous time and continuous space variables are defined constructively by establishing new convergence estimates for Markov chains on simplicial sequences. We emphasize high precision computability by numerical linear algebra methods as opposed to the ability of arriving to analytically closed form expressions in terms of special functions. Path dependent processes adapted to a given Markov filtration are associated to an operator algebra. If this algebra is commutative, the corresponding process is named Abelian, a concept which provides a far reaching extension of the notion of stochastic integral. We recover the classic Cameron-Dyson-Feynman-Girsanov-Ito-Kac-Martin theorem as a particular case of a broadly general block-diagonalization algorithm. This technique has many applications ranging from the problem of pricing cliquets to target-redemption-notes and volatility derivatives. Non-Abelian processes are also relevant and appear in several important applications to for instance snowballs and soft calls. We show that in these cases one can eectively use block-factorization algorithms. Finally, we discuss the method of dynamic conditioning that allows one to dynamically correlate over possibly even hundreds of processes in a numerically noiseless framework while preserving marginal distributions

OPERATOR METHODS, ABELIAN PROCESSES AND DYNAMIC CONDITIONING

A mathematical framework for Continuous Time Finance based on operator algebraic methods oers a new direct and entirely constructive perspective on the field. It also leads to new numerical analysis techniques which can take advantage of the emerging massively parallel GPU architectures which are uniquely suited to execute large matrix manipulations. This is partly a review paper as it covers and expands on the mathematical framework underlying a series of more applied articles. In addition, this article also presents a few key new theorems that make the treatment self-contained. Stochastic processes with continuous time and continuous space variables are defined constructively by establishing new convergence estimates for Markov chains on simplicial sequences. We emphasize high precision computability by numerical linear algebra methods as opposed to the ability of arriving to analytically closed form expressions in terms of special functions. Path dependent processes adapted to a given Markov filtration are associated to an operator algebra. If this algebra is commutative, the corresponding process is named Abelian, a concept which provides a far reaching extension of the notion of stochastic integral. We recover the classic Cameron-Dyson-Feynman-Girsanov-Ito-Kac-Martin theorem as a particular case of a broadly general block-diagonalization algorithm. This technique has many applications ranging from the problem of pricing cliquets to target-redemption-notes and volatility derivatives. Non-Abelian processes are also relevant and appear in several important applications to for instance snowballs and soft calls. We show that in these cases one can eectively use block-factorization algorithms. Finally, we discuss the method of dynamic conditioning that allows one to dynamically correlate over possibly even hundreds of processes in a numerically noiseless framework while preserving marginal distributions.Operator methods; financial derivatives; path-dependent derivatives; correlation derivatives

Fast Quantum Fourier Transforms for a Class of Non-abelian Groups

An algorithm is presented allowing the construction of fast Fourier transforms for any solvable group on a classical computer. The special structure of the recursion formula being the core of this algorithm makes it a good starting point to obtain systematically fast Fourier transforms for solvable groups on a quantum computer. The inherent structure of the Hilbert space imposed by the qubit architecture suggests to consider groups of order 2^n first (where n is the number of qubits). As an example, fast quantum Fourier transforms for all 4 classes of non-abelian 2-groups with cyclic normal subgroup of index 2 are explicitly constructed in terms of quantum circuits. The (quantum) complexity of the Fourier transform for these groups of size 2^n is O(n^2) in all cases.Comment: 16 pages, LaTeX2

Quantum algorithms for hidden nonlinear structures

Attempts to find new quantum algorithms that outperform classical computation have focused primarily on the nonabelian hidden subgroup problem, which generalizes the central problem solved by Shor's factoring algorithm. We suggest an alternative generalization, namely to problems of finding hidden nonlinear structures over finite fields. We give examples of two such problems that can be solved efficiently by a quantum computer, but not by a classical computer. We also give some positive results on the quantum query complexity of finding hidden nonlinear structures.Comment: 13 page

ACCAMS: Additive Co-Clustering to Approximate Matrices Succinctly

Matrix completion and approximation are popular tools to capture a user's preferences for recommendation and to approximate missing data. Instead of using low-rank factorization we take a drastically different approach, based on the simple insight that an additive model of co-clusterings allows one to approximate matrices efficiently. This allows us to build a concise model that, per bit of model learned, significantly beats all factorization approaches to matrix approximation. Even more surprisingly, we find that summing over small co-clusterings is more effective in modeling matrices than classic co-clustering, which uses just one large partitioning of the matrix. Following Occam's razor principle suggests that the simple structure induced by our model better captures the latent preferences and decision making processes present in the real world than classic co-clustering or matrix factorization. We provide an iterative minimization algorithm, a collapsed Gibbs sampler, theoretical guarantees for matrix approximation, and excellent empirical evidence for the efficacy of our approach. We achieve state-of-the-art results on the Netflix problem with a fraction of the model complexity.Comment: 22 pages, under review for conference publicatio