Applications of Riemannian optimization

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Riemannian joint dimensionality reduction and dictionary learning on symmetric positive definite

Dictionary learning (DL) and dimensionality reduction (DR) are powerful tools to analyze high-dimensional noisy signals. This paper proposes a novel Riemannian joint dimensionality reduction and dictionary learning (R-JDRDL) on symmetric positive definite (SPD) manifolds for classification tasks. Joint learning considers the interaction between dimensionality reduction and dictionary learning procedures by connecting them into a unified framework. We exploit a Riemannian optimization framework for solving DL and DR problems jointly.  Finally, we demonstrate that the proposed Riemannian Joint framework of Dimensionality Reduction and Dictionary Learning, dubbed R-JDRDL, outperforms existing state-of-the-art algorithms when used for image classification tasks.

Given {\bf X} \in \prod^N \mathcal{S}^m_{++} (m>d), we calculate \hat{{\bf U}}\in {\rm St}(d, m), \hat{{\bf D}}\in \prod^n \mathcal{S}^d_{++}, and \hat{{\bf A}}\in \mathbb{R}^{n\times N}_+(\geq 0) as            


 \displaystyle{ \min_{\scriptsize {\bf U}, {\bf D}, {\bf A} } \overbrace{J_d({\bf U}, {\bf D}, {\bf A})}^{\text{Discriminative} \atop \text{reconstruction error}} + \overbrace{\lambda_a J_a({\bf A})}^{\text{Graph-based coding} \atop \text{coefficient constraint}} + \underbrace{\lambda_u J_u({\bf U})}_{\text{Graph-based}\atop \text{projection constraint}} +  \underbrace{\lambda_1 R_s({\bf A})}_{\text{\L{1}-regularizer on} \atop \text{coding coefficient}} +\underbrace{\lambda_2 R_r({\bf A})}_{\text{\L{2}-regularizer on} \atop \text{coding coefficient}}+ \lambda_d R_d({\bf D}), }

where R_s({\bf A})={\bf 1}_n^T  |{\bf A} | {\bf 1}_N\ (:=\sum_{k=1}^K \sum_{i=1}^{I_k} \| {\bf a}_{k,i} \|_1),  R_r(\mat{A})=\| {\bf A} \|^2_F, and \lambda > 0.



  • HK and B.Mishra “Riemannian joint dimensionality reduction and dictionary learning on symmetric positive definite,” EUSIPCO, 2018.

Fast optimization algorithm on complex oblique manifold for hybrid precoding in Millimeter Wave MIMO systems

Hybrid precoding(-beamforming) is the most promising approach to reducing high hardware costs and high power consumption in large-scale millimeter wave (mmWave) MIMO systems. Hybrid precoding combines large-dimensional analog precoding (or beamforming) via phase shifters with lower-dimensional digital baseband precoding.

A maximization problem of spectral efficiency approximately boils down to a minimization problem of the Euclidean distance between the fully digital precoder and the hybrid precoder. This problem is further formulated as a matrix factorization problem of the fully digital precoder with a product of the digital baseband precoder matrix and the analog radio frequency (RF) precoder (or beamforming) matrix. The noteworthy point is that the phase shifters impose additional element-wise unit modulus constraints on the analog RF precoder matrix.

The conventional works suffer from performance loss in spectral efficiency, and also suffer from high complexities of optimization algorithms in terms of iteration as well as processing time. This paper proposes a new fast optimization algorithm that improves the calculation algorithm of the gradient and avoids the nested-loop architecture of the state-of-the-art algorithm, MO-AltMin. The problem is newly defined as

\displaystyle{\min_{\scriptsize {\rm vec}({\bf F}_{\text{RF}}) \in \mathcal{OB}(m, \mathbb{C})}  \| {\bf F}_{\text{OPT}} - {\bf F}_{\text{RF}} \hat{\bf F}_{\text{BB}} \|_F^2,}.

\displaystyle{\text{\ \ where\ \ } \hat{\bf F}_{\text{BB}} = {\bf F}_{\text{RF}}^{\dagger} {\bf F}_{\text{OPT}}.

Exhaustive evaluations demonstrate that the proposed algorithm yields comparable or faster processing speed than the state-of-the-art algorithms while keeping spectral efficiency near-optimal.

ミリ波MIMOシステムにおけるハイブリッド・プリコーディングの設計問題に着目し,スペクトル効率の向上を目指す高速最適化手法を提案する.当該問題は,一般に,フルディジタル・プリコーダ行列 {\bf F}_{\text{OPT}} とハイブリッド・プリコーダ行列間の距離の最小化問題として定式化される.ここでは,特に,ディジタル・ベースバンド・プリコーダ行列 {\bf F}_{\text{BB}} とアナログRFプリコーダ(ビーム・フォーミング)行列 {\bf F}_{\text{RF}} を用いた {\bf F}_{\text{OPT}} の行列分解 によるアプローチに着目する.当該問題において従来の複素Oblique多様体 上の最適化手法が直面する高い処理複雑さの問題に対して,繰り返し処理構造の最適化 による収束速度向上と,ベクトル化による大規模行列処理の回避 による処理量削減に基づく高速化手法を提案する.スペクトル効率と処理量に関するシミューレショーン実験から,準最適手法を含む最先端の手法に対する提案手法の優位性を示す.

Diagram of hybrid precoder.


  • HK, “Optimization algorithm on complex oblique manifold for hybrid precoding in Millimeter Wave MIMO systems,” IEEE GlobalSIP, 2018.

The software is available at Github.