Structural Loss Metrics for Tensor Approximation via Matrix Low-Rank Approximation

Matricized low-rank approximation via SVD is a standard surrogate for tensor decompositions, but entry-wise reconstruction error fails to capture multiway geometric degradation. Under an orthogonal Tucker model, we characterize this degradation using two metrics: cross-mode Direction Loss, measuring geometric subspace deviation from rank truncation and noise rotation, and Interaction Loss, quantifying multilinear interaction distortion in the core tensor. We prove that squared relative reconstruction error orthogonally decomposes into interaction loss and out-of-subspace energy, and derive a Wedin-type bound establishing the stability of a plug-in Direction Loss estimator. Experiments on synthetic and hyperspectral datasets demonstrate that nearly identical reconstruction errors can yield markedly different structural-loss profiles; hyperspectral patches with comparable reconstruction errors exhibit up to a 4.6-fold difference in Direction Loss, correlating with severe visual blurring.

Qualitative Evaluation of Language Model Rescoring in Automatic Speech Recognition

Evaluating automatic speech recognition (ASR) systems is a classical but difficult and still open problem, which often boils down to focusing only on the word error rate (WER). However, this metric suffers from many limitations and does not allow an in-depth analysis of automatic transcription errors. In this paper, we propose to study and understand the impact of rescoring using language models in ASR systems by means of several metrics often used in other natural language processing (NLP) tasks in addition to the WER. In particular, we introduce two measures related to morpho-syntactic and semantic aspects of transcribed words: 1) the POSER (Part-of-speech Error Rate), which should highlight the grammatical aspects, and 2) the EmbER (Embedding Error Rate), a measurement that modifies the WER by providing a weighting according to the semantic distance of the wrongly transcribed words. These metrics illustrate the linguistic contributions of the language models that are applied during a posterior rescoring step on transcription hypotheses.