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Let $G$ be a reductive algebraic group over an algebraically closed field $k$ of prime characteristic not $2$, whose Lie algebra is denoted $\mathfrak{g}$. We call a subvariety $\mathfrak{X}$ of the nilpotent cone $N \subset \mathfrak{g}$ monogamous if for every $e\in \mathfrak{X}$, the $\mathfrak{sl}_2$-triples $(e,h,f)$ with $f\in \mathfrak{X}$ are conjugate under the centraliser $C_G(e)$. Building on work by the first two authors, we show there is a unique maximal closed $G$-stable monogamous subvariety $V \subset N$ and that it is an orbit closure, hence irreducible. We show that $V$ can also be characterised in terms of Serre's $G$-complete reducibility., Comment: 15 pages

While neural network approaches have made significant strides in resolving classical signal processing problems, it is often the case that hybrid approaches that draw insight from both signal processing and neural networks produce more complete solutions. In this paper, we present a hybrid classical digital signal processing/deep neural network (DSP/DNN) approach to source separation (SS) highlighting the theoretical link between variational autoencoder and classical approaches to SS. We propose a system that transforms the single channel under-determined SS task to an equivalent multichannel over-determined SS problem in a properly designed latent space. The separation task in the latent space is treated as finding a variational block-wise disentangled representation of the mixture. We show empirically, that the design choices and the variational formulation of the task at hand motivated by the classical signal processing theoretical results lead to robustness to unseen out-of-distribution data and reduction of the overfitting risk. To address the resulting permutation issue we explicitly incorporate a novel differentiable permutation loss function and augment the model with a memory mechanism to keep track of the statistics of the individual sources.

We introduce a real-time, multichannel speech enhancement algorithm which maintains the spatial cues of stereo recordings including two speech sources. Recognizing that each source has unique spatial information, our method utilizes a dual-path structure, ensuring the spatial cues remain unaffected during enhancement by applying source-specific common-band gain. This method also seamlessly integrates pretrained monaural speech enhancement, eliminating the need for retraining on stereo inputs. Source separation from stereo mixtures is achieved via spatial beamforming, with the steering vector for each source being adaptively updated using post-enhancement output signal. This ensures accurate tracking of the spatial information. The final stereo output is derived by merging the spatial images of the enhanced sources, with its efficacy not heavily reliant on the separation performance of the beamforming. The algorithm runs in real-time on 10-ms frames with a 40 ms of look-ahead. Evaluations reveal its effectiveness in enhancing speech and preserving spatial cues in both fully and sparsely overlapped mixtures., Comment: Accepted for ICASSP 2024, 5 pages

A distinguished family of completely prime primitive ideals in the universal enveloping algebra of a reductive Lie algebra ${\mathfrak g}$ over ${\mathbb C}$ are those ideals constructed from one-dimensional representations of finite $W$-algebras. We refer to these ideals as Losev--Premet ideals. For ${\mathfrak g}$ simple of classical type, we prove that for a Losev-Premet ideal $I$ in $U({\mathfrak g})$, there exists a Losev-Premet ideal $I_0$ for a certain Levi subalgebra ${\mathfrak g}_0$ of ${\mathfrak g}$ such that associated variety of $I_0$ is the closure of a rigid nilpotent orbit in ${\mathfrak g}_0$ and $I$ is obtained from $I_0$ by parabolic induction. This is deduced from the corresponding statement about one-dimensional representations of finite $W$-algebras., Comment: 24 pages

Improving the interpretability of deep neural networks has recently gained increased attention, especially when the power of deep learning is leveraged to solve problems in physics. Interpretability helps us understand a model's ability to generalize and reveal its limitations. In this paper, we introduce a causal interpretable deep structure for modeling dynamic systems. Our proposed model makes use of the harmonic analysis by modeling the system in a time-frequency domain while maintaining high temporal and spectral resolution. Moreover, the model is built in an order recursive manner which allows for fast, robust, and exact second order optimization without the need for an explicit Hessian calculation. To circumvent the resulting high dimensionality of the building blocks of our system, a neural network is designed to identify the frequency interdependencies. The proposed model is illustrated and validated on nonlinear system identification problems as required for audio signal processing tasks. Crowd-sourced experimentation contrasting the performance of the proposed approach to other state-of-the-art solutions on an acoustic echo cancellation scenario confirms the effectiveness of our method for real-life applications.

Speech codec enhancement methods are designed to remove distortions added by speech codecs. While classical methods are very low in complexity and add zero delay, their effectiveness is rather limited. Compared to that, DNN-based methods deliver higher quality but they are typically high in complexity and/or require delay. The recently proposed Linear Adaptive Coding Enhancer (LACE) addresses this problem by combining DNNs with classical long-term/short-term postfiltering resulting in a causal low-complexity model. A short-coming of the LACE model is, however, that quality quickly saturates when the model size is scaled up. To mitigate this problem, we propose a novel adatpive temporal shaping module that adds high temporal resolution to the LACE model resulting in the Non-Linear Adaptive Coding Enhancer (NoLACE). We adapt NoLACE to enhance the Opus codec and show that NoLACE significantly outperforms both the Opus baseline and an enlarged LACE model at 6, 9 and 12 kb/s. We also show that LACE and NoLACE are well-behaved when used with an ASR system., Comment: final version, accepted at ICASSP 2024

In this study, we present an approach to train a single speech enhancement network that can perform both personalized and non-personalized speech enhancement. This is achieved by incorporating a frame-wise conditioning input that specifies the type of enhancement output. To improve the quality of the enhanced output and mitigate oversuppression, we experiment with re-weighting frames by the presence or absence of speech activity and applying augmentations to speaker embeddings. By training under a multi-task learning setting, we empirically show that the proposed unified model obtains promising results on both personalized and non-personalized speech enhancement benchmarks and reaches similar performance to models that are trained specialized for either task. The strong performance of the proposed method demonstrates that the unified model is a more economical alternative compared to keeping separate task-specific models during inference., Comment: Accepted by ICASSP 2023

Singing voice separation aims to separate music into vocals and accompaniment components. One of the major constraints for the task is the limited amount of training data with separated vocals. Data augmentation techniques such as random source mixing have been shown to make better use of existing data and mildly improve model performance. We propose a novel data augmentation technique, chromagram-based pitch-aware remixing, where music segments with high pitch alignment are mixed. By performing controlled experiments in both supervised and semi-supervised settings, we demonstrate that training models with pitch-aware remixing significantly improves the test signal-to-distortion ratio (SDR), Comment: To appear at ICASSP 2022, 5 pages, 1 figure

Let $k$ be an algebraically closed field of characteristic $p > 2$, let $n \in \mathbb Z_{>0}$, and take $G$ to be one of the classical algebraic groups $\mathrm{GL}_n(k)$, $\mathrm{SL}_n(k)$, $\mathrm{Sp}_n(k)$, $\mathrm O_n(k)$ or $\mathrm{SO}_n(k)$, with $\mathfrak g = \operatorname{Lie} G$. We determine the maximal $G$-stable closed subvariety $\mathcal V$ of the nilpotent cone $\mathcal N$ of $\mathfrak g$ such that the $G$-orbits in $\mathcal V$ are in bijection with the $G$-orbits of $\mathfrak{sl}_2$-triples $(e,h,f)$ with $e,f \in \mathcal V$. This result determines to what extent the theorems of Jacobson--Morozov and Kostant on $\mathfrak{sl}_2$-triples hold for classical algebraic groups over an algebraically closed field of "small" odd characteristic., Comment: 20 pages, minor changes

Current management of bipolar disorder relies on self-reported questionnaires and interviews with clinicians. The development of objective measures of deteriorating mood may also allow for early interventions to take place to avoid transitions into depressive states. The objective of this study was to use acceleration data recorded from smartphones to predict levels of depression in a population of participants diagnosed with bipolar disorder. Data were collected from 52 participants, with a mean of 37 weeks of acceleration data with a corresponding depression score recorded per participant. Time varying hidden Markov models were used to extract weekly features of activity, sleep and circadian rhythms. Personalised regression achieved mean absolute errors of 1.00(0.57) from a possible scale of 0 to 27 and was able to classify depression with an accuracy of 0.84(0.16). The results demonstrate features derived from smartphone accelerometers are able to provide objective markers of depression. Low barriers for uptake exist due to the widespread use of smartphones, with personalised models able to account for differences in the behaviour of individuals and provide accurate predictions of depression., Comment: 8 pages, 3 figures

We study the truncated shifted Yangian $Y_{n,l}(\sigma)$ over an algebraically closed field $\mathbb{k}$ of characteristic $p > 0$, which is known to be isomorphic to the finite $W$-algebra $U(\mathfrak{g}, e)$ associated to a corresponding nilpotent element $e\in \mathfrak{g} = \mathfrak{gl}_N(\mathbb{k})$. We obtain an explicit description of the centre of $Y_{n,l}(\sigma)$, showing that it is generated by its Harish-Chandra centre and its $p$-centre. We define $Y_{n,l}^{[p]}(\sigma)$ to be the quotient of $Y_{n,l}(\sigma)$ by the ideal generated by the kernel of trivial character of its $p$-centre. Our main theorem states that $Y_{n,l}^{[p]}(\sigma)$ is isomorphic to the restricted finite $W$-algebra $U^{[p]}(\mathfrak{g},e)$. As a consequence we obtain an explicit presentation of this restricted $W$-algebra., Comment: 38 pages. Comments welcome

Let $\mathfrak g = \mathfrak{gl}_N(k)$, where $k$ is an algebraically closed field of characteristic $p > 0$, and $N \in \mathbb Z_{\ge 1}$. Let $\chi \in \mathfrak g^*$ and denote by $U_\chi(\mathfrak g)$ the corresponding reduced enveloping algebra. The Kac--Weisfeiler conjecture, which was proved by Premet, asserts that any finite dimensional $U_\chi(\mathfrak g)$-module has dimension divisible by $p^{d_\chi}$, where $d_\chi$ is half the dimension of the coadjoint orbit of $\chi$. Our main theorem gives a classification of $U_\chi(\mathfrak g)$-modules of dimension $p^{d_\chi}$. As a consequence, we deduce that they are all parabolically induced from a 1-dimensional module for $U_0(\mathfrak h)$ for a certain Levi subalgebra $\mathfrak h$ of $\mathfrak g$; we view this as a modular analogue of M{\oe}glin's theorem on completely primitive ideals in $U(\mathfrak{gl}_N(\mathbb C))$. To obtain these results, we reduce to the case $\chi$ is nilpotent, and then classify the 1-dimensional modules for the corresponding restricted $W$-algebra., Comment: 24 pages, minor changes, to appear in Compositio Mathematica

Let $\mathfrak g$ be a simple Lie algebra over $\mathbb C$ and let $e \in \mathfrak g$ be nilpotent. We consider the finite $W$-algebra $U(\mathfrak g,e)$ associated to $e$ and the problem of determining the variety $\mathcal E(\mathfrak g,e)$ of 1-dimensional representations of $U(\mathfrak g,e)$. For $\mathfrak g$ of low rank, we report on computer calculations that have been used to determine the structure of $\mathcal E(\mathfrak g,e)$, and the action of the component group $\Gamma_e$ of the centralizer of $e$ on $\mathcal E(\mathfrak g,e)$. As a consequence, we provide two examples where the nilpotent orbit of $e$ is induced, but there is a 1-dimensional $\Gamma_e$-stable $U(\mathfrak g,e)$-module which is not induced via Losev's parabolic induction functor. In turn this gives examples where there is a "non-induced" multiplicity free primitive ideal of $U(\mathfrak g)$., Comment: 13 pages

Let $k$ be an algebraically closed field of characteristic $p > 0$ and let $G$ be a connected reductive algebraic group over $k$. Under some standard hypothesis on $G$, we give a direct approach to the finite $W$-algebra $U(\mathfrak g,e)$ associated to a nilpotent element $e \in \mathfrak g = \operatorname{Lie} G$. We prove a PBW theorem and deduce a number of consequences, then move on to define and study the $p$-centre of $U(\mathfrak g,e)$, which allows us to define reduced finite $W$-algebras $U_\eta(\mathfrak g,e)$ and we verify that they coincide with those previously appearing in the work of Premet. Finally, we prove a modular version of Skryabin's equivalence of categories, generalizing recent work of the second author., Comment: 31 pages, minor changes

Let $W_{m|n}$ be the (finite) $W$-algebra attached to the principal nilpotent orbit in the general linear Lie superalgebra $\mathfrak{gl}_{m|n}(\mathbb{C})$. In this paper we study the {\em Whittaker coinvariants functor}, which is an exact functor from category $\mathcal O$ for $\mathfrak{gl}_{m|n}(\mathbb{C})$ to a certain category of finite-dimensional modules over $W_{m|n}$. We show that this functor has properties similar to Soergel's functor $\mathbb V$ in the setting of category $\mathcal O$ for a semisimple Lie algebra. We also use it to compute the center of $W_{m|n}$ explicitly, and deduce some consequences for the classification of blocks of $\mathcal O$ up to Morita/derived equivalence., Comment: 58 pages, minor changes