Numerically stable coded matrix computations via circulant and rotation matrix embeddings

Several recent works have used coding-theoretic ideas for mitigating the effect of stragglers in distributed matrix computations (matrix-vector and matrix-matrix multiplication) over the reals. In particular, a polynomial code based approach distributes matrix-matrix multiplication among n worker nodes by means of polynomial evaluations. This allows for an ``optimal\u27\u27 recovery threshold whereby the intended result can be decoded as long as at least (n−s) worker nodes complete their tasks; s is the number of stragglers that the scheme can handle. However, a major issue with these approaches is the high condition number of the corresponding Vandermonde-structured recovery matrices. This presents serious numerical precision issues when decoding the desired result. It is well known that the condition number of real Vandermonde matrices grows exponentially in n. In contrast, the condition numbers of Vandermonde matrices with parameters on the unit circle are much better behaved. In this work we leverage the properties of circulant permutation matrices and rotation matrices to obtain coded computation schemes with significantly lower worst case condition numbers; these matrices have eigenvalues that lie on the unit circle. Our scheme is such that the associated recovery matrices have a condition number corresponding to Vandermonde matrices with parameters given by the eigenvalues of the corresponding circulant permutation and rotation matrices. We demonstrate an upper bound on the worst case condition number of these matrices which grows as ≈O(ns+6). In essence, we leverage the well-behaved conditioning of complex Vandermonde matrices with parameters on the unit circle, while still working with computation over the reals. Experimental results demonstrate that our proposed method has condition numbers that are several orders of magnitude better than prior work

Irreversibility in quantum maps with decoherence

The Bolztmann echo (BE) is a measure of irreversibility and sensitivity to perturbations for non-isolated systems. Recently, different regimes of this quantity were described for chaotic systems. There is a perturbative regime where the BE decays with a rate given by the sum of a term depending on the accuracy with which the system is time-reversed and a term depending on the coupling between the system and the environment. In addition, a parameter independent regime, characterised by the classical Lyapunov exponent, is expected. In this paper we study the behaviour of the BE in hyperbolic maps that are in contact with different environments. We analyse the emergence of the different regimes and show that the behaviour of the decay rate of the BE is strongly dependent on the type of environment.Comment: 13 pages, 3 figures

Coded matrix computation with gradient coding

Polynomial based approaches, such as the Mat-Dot and entangled polynomial (EP) codes have been used extensively within coded matrix computations to obtain schemes with good thresholds. However, these schemes are well-recognized to suffer from poor numerical stability in decoding. Moreover, the encoding process in these schemes involves linearly combining a large number of input submatrices, i.e., the encoding weight is high. For the practically relevant case of sparse input matrices, this can have the undesirable effect of significantly increasing the worker node computation time. In this work, we propose a generalization of the EP scheme by combining the idea of gradient coding along with the basic EP encoding. Our scheme allows us to reduce the weight of the encoding and arrive at schemes that exhibit much better numerical stability; this is achieved at the expense of a worse threshold. By appropriately setting parameters in our scheme, we recover several well-known schemes in the literature. Simulation results show that our scheme provides excellent numerical stability and fast computation speed (for sparse input matrices) as compared to EPC and Mat-Dot codes