Complex Interpolation of Matrices with an application to Multi-Manifold Learning

Given two symmetric positive-definite matrices $A, B \in \mathbb{R}^{n \times n}$, we study the spectral properties of the interpolation $A^{1-x} B^x$ for $0 \leq x \leq 1$. The presence of `common structures' in $A$ and $B$, eigenvectors pointing in a similar direction, can be investigated using this interpolation perspective. Generically, exact log-linearity of the operator norm $\|A^{1-x} B^x\|$ is equivalent to the existence of a shared eigenvector in the original matrices; stability bounds show that approximate log-linearity forces principal singular vectors to align with leading eigenvectors of both matrices. These results give rise to and provide theoretical justification for a multi-manifold learning framework that identifies common and distinct latent structures in multiview data.

DiT-Flow: Speech Enhancement Robust to Multiple Distortions based on Flow Matching in Latent Space and Diffusion Transformers

Recent advances in generative models, such as diffusion and flow matching, have shown strong performance in audio tasks. However, speech enhancement (SE) models are typically trained on limited datasets and evaluated under narrow conditions, limiting real-world applicability. To address this, we propose DiT-Flow, a flow matching-based SE framework built on the latent Diffusion Transformer (DiT) backbone and trained for robustness across diverse distortions, including noise, reverberation, and compression. DiT-Flow operates on compact variational auto-encoders (VAEs)-derived latent features. We validated our approach on StillSonicSet, a synthetic yet acoustically realistic dataset composed of LibriSpeech, FSD50K, FMA, and 90 Matterport3D scenes. Experiments show that DiT-Flow consistently outperforms state-of-the-art generative SE models, demonstrating the effectiveness of flow matching in multi-condition speech enhancement. Despite ongoing efforts to expand synthetic data realism, a persistent bottleneck in SE is the inevitable mismatch between training and deployment conditions. By integrating LoRA with the MoE framework, we achieve both parameter-efficient and high-performance training for DiT-Flow robust to multiple distortions with using 4.9% percentage of the total parameters to obtain a better performance on five unseen distortions.