[Paper Review] Estimating Certain Non-Zero Littlewood-Richardson Coefficients

Hariharan Narayanan|arXiv (Cornell University)|Jun 18, 2013

Advanced Combinatorial Mathematics13 references3 citations

TL;DR

This paper presents a randomized strongly polynomial-time approximation scheme for estimating certain non-zero Littlewood-Richardson coefficients, leveraging volume estimation via the Dikin walk and Brunn-Minkowski-type inequalities. It establishes that a constant fraction of coefficients in a shifted Littlewood-Richardson cone can be approximated efficiently, and proves a form of approximate log-concavity for these coefficients under specific conditions.

ABSTRACT

Littlewood Richardson coefficients are structure constants appearing in the representation theory of the general linear groups ($GL_n$). The main results of this paper are: 1. A strongly polynomial randomized approximation scheme for certain Littlewood-Richardson coefficients. 2. A proof of approximate log-concavity of certain Littlewood-Richardson coefficients.

Motivation & Objective

• To develop an efficient approximation scheme for Littlewood-Richardson coefficients, which are #P-hard to compute exactly in general.
• To address the challenge of approximating these coefficients when the rank of the general linear group is not fixed, where exact computation is infeasible under standard complexity assumptions.
• To establish approximate log-concavity properties of the coefficients in a specific region of the Littlewood-Richardson cone.
• To provide a randomized algorithm with strongly polynomial runtime that depends only on input size and error parameters, not on bit-length of input data.
• To extend the scope of efficient approximation beyond known positivity testing and fixed-rank cases to a large, dense subset of the coefficient space.

Proposed method

• The method uses volume estimation of polytopes $ Q_{ u}^{ u} $ associated with Littlewood-Richardson coefficients via the Dikin walk, a Markov chain Monte Carlo technique for sampling from log-concave distributions.
• It applies Hoeffding's inequality to bound the error in estimating the ratio of volumes $ \mathrm{vol}(\zeta_{\lambda\mu}^{\nu}) / \mathrm{vol}(Q_{\lambda\mu}^{\nu}) $, ensuring high-probability concentration.
• The algorithm operates in a real-number model with truncated real arithmetic, ensuring that bit-length growth does not affect the strongly polynomial runtime.
• It relies on the Brunn-Minkowski inequality to relate volumes of convex sets under Minkowski combinations, enabling the derivation of approximate log-concavity.
• The key technical step involves showing that for vectors in $ (1/\epsilon)(\Delta, \Delta, \Delta') + LRC $, the coefficient $ c_{\lambda\mu}^\nu $ is within a $ 1 \pm C\epsilon $ factor of the volume of the associated polytope.
• The construction uses a shift by $ (\Delta, \Delta, \Delta') $ to ensure that the coefficient estimates are well-approximated by volume ratios, enabling efficient sampling and estimation.

Experimental results

Research questions

• RQ1 Can a randomized strongly polynomial-time algorithm approximate a large fraction of non-zero Littlewood-Richardson coefficients with high probability and desired accuracy?
• RQ2 Is there a form of approximate log-concavity that holds for Littlewood-Richardson coefficients in a specific region of the cone, despite the failure of exact log-concavity?
• RQ3 What fraction of all Littlewood-Richardson coefficients can be efficiently approximated when the input size is bounded by $ \gamma $, and how does this fraction depend on $ n $ and $ \gamma $?
• RQ4 How does the volume of the polytope $ Q_{\lambda\mu}^\nu $ relate to the actual coefficient $ c_{\lambda\mu}^\nu $ in the shifted cone $ (1/\epsilon)(\Delta, \Delta, \Delta') + LRC $?
• RQ5 Can the Dikin walk be adapted to provide efficient sampling from the uniform measure on $ Q_{\lambda\mu}^\nu $ with bounded error, even when working with truncated real numbers?

Key findings

• A randomized strongly polynomial-time approximation scheme exists for Littlewood-Richardson coefficients in the region $ (\Delta, \Delta, \Delta') + LRC $, with runtime polynomial in $ n $, $ \epsilon^{-1} $, and $ \log \delta^{-1} $, independent of input bit-length.
• For $ \gamma > Cn^5 $, a $ 1 - C(n^5 / \gamma) $ fraction of all non-zero coefficients in $ LRC \cap \{ \| (\lambda,\mu,\nu) \|_1 \leq \gamma \} $ can be approximated within $ 1 \pm C\epsilon $ with high probability.
• The coefficients in $ (1/\epsilon)(\Delta, \Delta, \Delta') + LRC $ satisfy an approximate log-concavity inequality: $ \log c_{\lambda\mu}^\nu + C\epsilon \geq \theta \log c_{\lambda'\mu'}^{\nu'} + (1-\theta) \log c_{\bar{\lambda}\bar{\mu}}^{\bar{\nu}} $, for $ n > C $.
• The ratio $ c_{\lambda\mu}^\nu / \mathrm{vol}(Q_{\lambda\mu}^\nu) $ lies within $ [1 - C\epsilon, 1] $ for all $ (\lambda,\mu,\nu) \in (1/\epsilon)(\Delta, \Delta, \Delta') + LRC $, ensuring volume estimation approximates the coefficient well.
• The success probability of the volume estimation procedure is bounded below by $ 1 - \delta $, with the error probability controlled via Hoeffding's inequality and the number of samples polynomial in $ \epsilon^{-1} $ and $ \log \delta^{-1} $.
• The method achieves a multiplicative approximation factor of $ 1 \pm C\epsilon $, with the error bound independent of the input size in bit-length, confirming the strongly polynomial nature of the algorithm.