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Diffusion models have found great success in generating high quality, natural samples of speech, but their potential for density estimation for speech has so far remained largely unexplored. In this work, we leverage an unconditional diffusion model trained only on clean speech for the assessment of speech quality. We show that the quality of a speech utterance can be assessed by estimating the likelihood of a corresponding sample in the terminating Gaussian distribution, obtained via a deterministic noising process. The resulting method is purely unsupervised, trained only on clean speech, and therefore does not rely on annotations. Our diffusion-based approach leverages clean speech priors to assess quality based on how the input relates to the learned distribution of clean data. Our proposed log-likelihoods show promising results, correlating well with intrusive speech quality metrics such as POLQA and SI-SDR.

We present a head-related transfer function (HRTF) estimation method which relies on a data-driven prior given by a score-based diffusion model. The HRTF is estimated in reverberant environments using natural excitation signals, e.g. human speech. The impulse response of the room is estimated along with the HRTF by optimizing a parametric model of reverberation based on the statistical behaviour of room acoustics. The posterior distribution of HRTF given the reverberant measurement and excitation signal is modelled using the score-based HRTF prior and a log-likelihood approximation. We show that the resulting method outperforms several baselines, including an oracle recommender system that assigns the optimal HRTF in our training set based on the smallest distance to the true HRTF at the given direction of arrival. In particular, we show that the diffusion prior can account for the large variability of high-frequency content in HRTFs.

This paper presents an unsupervised method for single-channel blind dereverberation and room impulse response (RIR) estimation, called BUDDy. The algorithm is rooted in Bayesian posterior sampling: it combines a likelihood model enforcing fidelity to the reverberant measurement, and an anechoic speech prior implemented by an unconditional diffusion model. We design a parametric filter representing the RIR, with exponential decay for each frequency subband. Room acoustics estimation and speech dereverberation are jointly carried out, as the filter parameters are iteratively estimated and the speech utterance refined along the reverse diffusion trajectory. In a blind scenario where the room impulse response is unknown, BUDDy successfully performs speech dereverberation in various acoustic scenarios, significantly outperforming other blind unsupervised baselines. Unlike supervised methods, which often struggle to generalize, BUDDy seamlessly adapts to different acoustic conditions. This paper extends our previous work by offering new experimental results and insights into the algorithm's performance and versatility. We first investigate the robustness of informed dereverberation methods to RIR estimation errors, to motivate the joint acoustic estimation and dereverberation paradigm. Then, we demonstrate the adaptability of our method to high-resolution singing voice dereverberation, study its performance in RIR estimation, and conduct subjective evaluation experiments to validate the perceptual quality of the results, among other contributions. Audio samples and code can be found online., Comment: Submitted to IEEE/ACM Transactions on Audio, Speech and Language Processing

Music generation schemes using language modeling rely on a vocabulary of audio tokens, generally provided as codes in a discrete latent space learnt by an auto-encoder. Multi-stage quantizers are often employed to produce these tokens, therefore the decoding strategy used for token prediction must be adapted to account for multiple codebooks: either it should model the joint distribution over all codebooks, or fit the product of the codebook marginal distributions. Modelling the joint distribution requires a costly increase in the number of auto-regressive steps, while fitting the product of the marginals yields an inexact model unless the codebooks are mutually independent. In this work, we introduce an independence-promoting loss to regularize the auto-encoder used as the tokenizer in language models for music generation. The proposed loss is a proxy for mutual information based on the maximum mean discrepancy principle, applied in reproducible kernel Hilbert spaces. Our criterion is simple to implement and train, and it is generalizable to other multi-stream codecs. We show that it reduces the statistical dependence between codebooks during auto-encoding. This leads to an increase in the generated music quality when modelling the product of the marginal distributions, while generating audio much faster than the joint distribution model., Comment: Accepted to ICML 2024

In this paper, we present an unsupervised single-channel method for joint blind dereverberation and room impulse response estimation, based on posterior sampling with diffusion models. We parameterize the reverberation operator using a filter with exponential decay for each frequency subband, and iteratively estimate the corresponding parameters as the speech utterance gets refined along the reverse diffusion trajectory. A measurement consistency criterion enforces the fidelity of the generated speech with the reverberant measurement, while an unconditional diffusion model implements a strong prior for clean speech generation. Without any knowledge of the room impulse response nor any coupled reverberant-anechoic data, we can successfully perform dereverberation in various acoustic scenarios. Our method significantly outperforms previous blind unsupervised baselines, and we demonstrate its increased robustness to unseen acoustic conditions in comparison to blind supervised methods. Audio samples and code are available online., Comment: Submitted to IWAENC 2024

Signature kernels are at the core of several machine learning algorithms for analysing multivariate time series. The kernel of two bounded variation paths (such as piecewise linear interpolations of time series data) is typically computed by solving a Goursat problem for a hyperbolic partial differential equation (PDE) in two independent time variables. However, this approach becomes considerably less practical for highly oscillatory input paths, as they have to be resolved at a fine enough scale to accurately recover their signature kernel, resulting in significant time and memory complexities. To mitigate this issue, we first show that the signature kernel of a broader class of paths, known as \emph{smooth rough paths}, also satisfies a PDE, albeit in the form of a system of coupled equations. We then use this result to introduce new algorithms for the numerical approximation of signature kernels. As bounded variation paths (and more generally geometric $p$-rough paths) can be approximated by piecewise smooth rough paths, one can replace the PDE with rapidly varying coefficients in the original Goursat problem by an explicit system of coupled equations with piecewise constant coefficients derived from the first few iterated integrals of the original input paths. While this approach requires solving more equations, they do not require looking back at the complex and fine structure of the initial paths, which significantly reduces the computational complexity associated with the analysis of highly oscillatory time series.

We introduce SigNova, a new semi-supervised framework for detecting anomalies in streamed data. While our initial examples focus on detecting radio-frequency interference (RFI) in digitized signals within the field of radio astronomy, it is important to note that SigNova's applicability extends to any type of streamed data. The framework comprises three primary components. Firstly, we use the signature transform to extract a canonical collection of summary statistics from observational sequences. This allows us to represent variable-length visibility samples as finite-dimensional feature vectors. Secondly, each feature vector is assigned a novelty score, calculated as the Mahalanobis distance to its nearest neighbor in an RFI-free training set. By thresholding these scores we identify observation ranges that deviate from the expected behavior of RFI-free visibility samples without relying on stringent distributional assumptions. Thirdly, we integrate this anomaly detector with Pysegments, a segmentation algorithm, to localize consecutive observations contaminated with RFI, if any. This approach provides a compelling alternative to classical windowing techniques commonly used for RFI detection. Importantly, the complexity of our algorithm depends on the RFI pattern rather than on the size of the observation window. We demonstrate how SigNova improves the detection of various types of RFI (e.g., broadband and narrowband) in time-frequency visibility data. We validate our framework on the Murchison Widefield Array (MWA) telescope and simulated data and the Hydrogen Epoch of Reionization Array (HERA).

With the development of audio playback devices and fast data transmission, the demand for high sound quality is rising for both entertainment and communications. In this quest for better sound quality, challenges emerge from distortions and interferences originating at the recording side or caused by an imperfect transmission pipeline. To address this problem, audio restoration methods aim to recover clean sound signals from the corrupted input data. We present here audio restoration algorithms based on diffusion models, with a focus on speech enhancement and music restoration tasks. Traditional approaches, often grounded in handcrafted rules and statistical heuristics, have shaped our understanding of audio signals. In the past decades, there has been a notable shift towards data-driven methods that exploit the modeling capabilities of DNNs. Deep generative models, and among them diffusion models, have emerged as powerful techniques for learning complex data distributions. However, relying solely on DNN-based learning approaches carries the risk of reducing interpretability, particularly when employing end-to-end models. Nonetheless, data-driven approaches allow more flexibility in comparison to statistical model-based frameworks, whose performance depends on distributional and statistical assumptions that can be difficult to guarantee. Here, we aim to show that diffusion models can combine the best of both worlds and offer the opportunity to design audio restoration algorithms with a good degree of interpretability and a remarkable performance in terms of sound quality. We explain the diffusion formalism and its application to the conditional generation of clean audio signals. We believe that diffusion models open an exciting field of research with the potential to spawn new audio restoration algorithms that are natural-sounding and remain robust in difficult acoustic situations., Comment: Currently in revision for IEEE Signal Processing Magazine Special Issue "Model-based and Data-Driven Audio Signal Processing"

Diffusion models have shown promising results in speech enhancement, using a task-adapted diffusion process for the conditional generation of clean speech given a noisy mixture. However, at test time, the neural network used for score estimation is called multiple times to solve the iterative reverse process. This results in a slow inference process and causes discretization errors that accumulate over the sampling trajectory. In this paper, we address these limitations through a two-stage training approach. In the first stage, we train the diffusion model the usual way using the generative denoising score matching loss. In the second stage, we compute the enhanced signal by solving the reverse process and compare the resulting estimate to the clean speech target using a predictive loss. We show that using this second training stage enables achieving the same performance as the baseline model using only 5 function evaluations instead of 60 function evaluations. While the performance of usual generative diffusion algorithms drops dramatically when lowering the number of function evaluations (NFEs) to obtain single-step diffusion, we show that our proposed method keeps a steady performance and therefore largely outperforms the diffusion baseline in this setting and also generalizes better than its predictive counterpart., Comment: copyright 2023 IEEE. Personal use of this material is permitted. Permission from IEEE must be obtained for all other uses, in any current or future media, including reprinting/republishing this material for advertising or promotional purposes, creating new collective works, for resale or redistribution to servers or lists, or reuse of any copyrighted component of this work in other works

In this paper we present a method for single-channel wind noise reduction using our previously proposed diffusion-based stochastic regeneration model combining predictive and generative modelling. We introduce a non-additive speech in noise model to account for the non-linear deformation of the membrane caused by the wind flow and possible clipping. We show that our stochastic regeneration model outperforms other neural-network-based wind noise reduction methods as well as purely predictive and generative models, on a dataset using simulated and real-recorded wind noise. We further show that the proposed method generalizes well by testing on an unseen dataset with real-recorded wind noise. Audio samples, data generation scripts and code for the proposed methods can be found online (https://uhh.de/inf-sp-storm-wind)., Comment: Accepted to VDE 15th ITG conference on Speech Communication

We present in this paper an informed single-channel dereverberation method based on conditional generation with diffusion models. With knowledge of the room impulse response, the anechoic utterance is generated via reverse diffusion using a measurement consistency criterion coupled with a neural network that represents the clean speech prior. The proposed approach is largely more robust to measurement noise compared to a state-of-the-art informed single-channel dereverberation method, especially for non-stationary noise. Furthermore, we compare to other blind dereverberation methods using diffusion models and show superiority of the proposed approach for large reverberation times. We motivate the presented algorithm by introducing an extension for blind dereverberation allowing joint estimation of the room impulse response and anechoic speech. Audio samples and code can be found online (https://uhh.de/inf-sp-derev-dps).

Since its inception, the field of deep speech enhancement has been dominated by predictive (discriminative) approaches, such as spectral mapping or masking. Recently, however, novel generative approaches have been applied to speech enhancement, attaining good denoising performance with high subjective quality scores. At the same time, advances in deep learning also allowed for the creation of neural network-based metrics, which have desirable traits such as being able to work without a reference (non-intrusively). Since generatively enhanced speech tends to exhibit radically different residual distortions, its evaluation using instrumental speech metrics may behave differently compared to predictively enhanced speech. In this paper, we evaluate the performance of the same speech enhancement backbone trained under predictive and generative paradigms on a variety of metrics and show that intrusive and non-intrusive measures correlate differently for each paradigm. This analysis motivates the search for metrics that can together paint a complete and unbiased picture of speech enhancement performance, irrespective of the model's training process., Comment: Submitted to ITG Conference on Speech Communication

Neural SDEs are continuous-time generative models for sequential data. State-of-the-art performance for irregular time series generation has been previously obtained by training these models adversarially as GANs. However, as typical for GAN architectures, training is notoriously unstable, often suffers from mode collapse, and requires specialised techniques such as weight clipping and gradient penalty to mitigate these issues. In this paper, we introduce a novel class of scoring rules on pathspace based on signature kernels and use them as objective for training Neural SDEs non-adversarially. By showing strict properness of such kernel scores and consistency of the corresponding estimators, we provide existence and uniqueness guarantees for the minimiser. With this formulation, evaluating the generator-discriminator pair amounts to solving a system of linear path-dependent PDEs which allows for memory-efficient adjoint-based backpropagation. Moreover, because the proposed kernel scores are well-defined for paths with values in infinite dimensional spaces of functions, our framework can be easily extended to generate spatiotemporal data. Our procedure permits conditioning on a rich variety of market conditions and significantly outperforms alternative ways of training Neural SDEs on a variety of tasks including the simulation of rough volatility models, the conditional probabilistic forecasts of real-world forex pairs where the conditioning variable is an observed past trajectory, and the mesh-free generation of limit order book dynamics., Comment: Code available at https://github.com/issaz/sigker-nsdes/

Distribution Regression on path-space refers to the task of learning functions mapping the law of a stochastic process to a scalar target. The learning procedure based on the notion of path-signature, i.e. a classical transform from rough path theory, was widely used to approximate weakly continuous functionals, such as the pricing functionals of path--dependent options' payoffs. However, this approach fails for Optimal Stopping Problems arising from mathematical finance, such as the pricing of American options, because the corresponding value functions are in general discontinuous with respect to the weak topology. In this paper we develop a rigorous mathematical framework to resolve this issue by recasting an Optimal Stopping Problem as a higher order kernel mean embedding regression based on the notions of higher rank signatures of measure--valued paths and adapted topologies. The core computational component of our algorithm consists in solving a family of two--dimensional hyperbolic PDEs., Comment: 33 pages

Motivated by the paradigm of reservoir computing, we consider randomly initialized controlled ResNets defined as Euler-discretizations of neural controlled differential equations (Neural CDEs), a unified architecture which enconpasses both RNNs and ResNets. We show that in the infinite-width-depth limit and under proper scaling, these architectures converge weakly to Gaussian processes indexed on some spaces of continuous paths and with kernels satisfying certain partial differential equations (PDEs) varying according to the choice of activation function, extending the results of Hayou (2022); Hayou & Yang (2023) to the controlled and homogeneous case. In the special, homogeneous, case where the activation is the identity, we show that the equation reduces to a linear PDE and the limiting kernel agrees with the signature kernel of Salvi et al. (2021a). We name this new family of limiting kernels neural signature kernels. Finally, we show that in the infinite-depth regime, finite-width controlled ResNets converge in distribution to Neural CDEs with random vector fields which, depending on whether the weights are shared across layers, are either time-independent and Gaussian or behave like a matrix-valued Brownian motion., Comment: Added commutativity of limits, ICML 2023 final version

In this paper, we present a causal speech signal improvement system that is designed to handle different types of distortions. The method is based on a generative diffusion model which has been shown to work well in scenarios with missing data and non-linear corruptions. To guarantee causal processing, we modify the network architecture of our previous work and replace global normalization with causal adaptive gain control. We generate diverse training data containing a broad range of distortions. This work was performed in the context of an "ICASSP Signal Processing Grand Challenge" and submitted to the non-real-time track of the "Speech Signal Improvement Challenge 2023", where it was ranked fifth., Comment: Accepted by ICASSP 2023

In this paper, we present a scheme for extending deep neural network-based multiplicative maskers to deep subband filters for speech restoration in the time-frequency domain. The resulting method can be generically applied to any deep neural network providing masks in the time-frequency domain, while requiring only few more trainable parameters and a computational overhead that is negligible for state-of-the-art neural networks. We demonstrate that the resulting deep subband filtering scheme outperforms multiplicative masking for dereverberation, while leaving the denoising performance virtually the same. We argue that this is because deep subband filtering in the time-frequency domain fits the subband approximation often assumed in the dereverberation literature, whereas multiplicative masking corresponds to the narrowband approximation generally employed for denoising., Comment: Accepted at ISCA Interspeech 2023

Diffusion models have shown a great ability at bridging the performance gap between predictive and generative approaches for speech enhancement. We have shown that they may even outperform their predictive counterparts for non-additive corruption types or when they are evaluated on mismatched conditions. However, diffusion models suffer from a high computational burden, mainly as they require to run a neural network for each reverse diffusion step, whereas predictive approaches only require one pass. As diffusion models are generative approaches they may also produce vocalizing and breathing artifacts in adverse conditions. In comparison, in such difficult scenarios, predictive models typically do not produce such artifacts but tend to distort the target speech instead, thereby degrading the speech quality. In this work, we present a stochastic regeneration approach where an estimate given by a predictive model is provided as a guide for further diffusion. We show that the proposed approach uses the predictive model to remove the vocalizing and breathing artifacts while producing very high quality samples thanks to the diffusion model, even in adverse conditions. We further show that this approach enables to use lighter sampling schemes with fewer diffusion steps without sacrificing quality, thus lifting the computational burden by an order of magnitude. Source code and audio examples are available online (https://uhh.de/inf-sp-storm)., Comment: Published in IEEE/ACM Transactions on Audio, Speech and Language Processing, 2023

Diffusion-based generative models have had a high impact on the computer vision and speech processing communities these past years. Besides data generation tasks, they have also been employed for data restoration tasks like speech enhancement and dereverberation. While discriminative models have traditionally been argued to be more powerful e.g. for speech enhancement, generative diffusion approaches have recently been shown to narrow this performance gap considerably. In this paper, we systematically compare the performance of generative diffusion models and discriminative approaches on different speech restoration tasks. For this, we extend our prior contributions on diffusion-based speech enhancement in the complex time-frequency domain to the task of bandwith extension. We then compare it to a discriminatively trained neural network with the same network architecture on three restoration tasks, namely speech denoising, dereverberation and bandwidth extension. We observe that the generative approach performs globally better than its discriminative counterpart on all tasks, with the strongest benefit for non-additive distortion models, like in dereverberation and bandwidth extension. Code and audio examples can be found online at https://uhh.de/inf-sp-sgmsemultitask, Comment: \c{opyright} 2023 IEEE. Personal use of this material is permitted. Permission from IEEE must be obtained for all other uses, in any current or future media, including reprinting/republishing this material for advertising or promotional purposes, creating new collective works, for resale or redistribution to servers or lists, or reuse of any copyrighted component of this work in other works

In this work, we build upon our previous publication and use diffusion-based generative models for speech enhancement. We present a detailed overview of the diffusion process that is based on a stochastic differential equation and delve into an extensive theoretical examination of its implications. Opposed to usual conditional generation tasks, we do not start the reverse process from pure Gaussian noise but from a mixture of noisy speech and Gaussian noise. This matches our forward process which moves from clean speech to noisy speech by including a drift term. We show that this procedure enables using only 30 diffusion steps to generate high-quality clean speech estimates. By adapting the network architecture, we are able to significantly improve the speech enhancement performance, indicating that the network, rather than the formalism, was the main limitation of our original approach. In an extensive cross-dataset evaluation, we show that the improved method can compete with recent discriminative models and achieves better generalization when evaluating on a different corpus than used for training. We complement the results with an instrumental evaluation using real-world noisy recordings and a listening experiment, in which our proposed method is rated best. Examining different sampler configurations for solving the reverse process allows us to balance the performance and computational speed of the proposed method. Moreover, we show that the proposed method is also suitable for dereverberation and thus not limited to additive background noise removal. Code and audio examples are available online, see https://github.com/sp-uhh/sgmse, Comment: Accepted version

A two-stage lightweight online dereverberation algorithm for hearing devices is presented in this paper. The approach combines a multi-channel multi-frame linear filter with a single-channel single-frame post-filter. Both components rely on power spectral density (PSD) estimates provided by deep neural networks (DNNs). By deriving new metrics analyzing the dereverberation performance in various time ranges, we confirm that directly optimizing for a criterion at the output of the multi-channel linear filtering stage results in a more efficient dereverberation as compared to placing the criterion at the output of the DNN to optimize the PSD estimation. More concretely, we show that training this stage end-to-end helps further remove the reverberation in the range accessible to the filter, thus increasing the \textit{early-to-moderate} reverberation ratio. We argue and demonstrate that it can then be well combined with a post-filtering stage to efficiently suppress the residual late reverberation, thereby increasing the \textit{early-to-final} reverberation ratio. This proposed two stage procedure is shown to be both very effective in terms of dereverberation performance and computational demands, as compared to e.g. recent state-of-the-art DNN approaches. Furthermore, the proposed two-stage system can be adapted to the needs of different types of hearing-device users by controlling the amount of reduction of early reflections., Comment: Accepted for publication in EURASIP Journal on Audio, Speech and Music Processing

In this paper, a neural network-augmented algorithm for noise-robust online dereverberation with a Kalman filtering variant of the weighted prediction error (WPE) method is proposed. The filter stochastic variations are predicted by a deep neural network (DNN) trained end-to-end using the filter residual error and signal characteristics. The presented framework allows for robust dereverberation on a single-channel noisy reverberant dataset similar to WHAMR!. The Kalman filtering WPE introduces distortions in the enhanced signal when predicting the filter variations from the residual error only, if the target speech power spectral density is not perfectly known and the observation is noisy. The proposed approach avoids these distortions by correcting the filter variations estimation in a data-driven way, increasing the robustness of the method to noisy scenarios. Furthermore, it yields a strong dereverberation and denoising performance compared to a DNN-supported recursive least squares variant of WPE, especially for highly noisy inputs., Comment: accepted to INTERSPEECH 2022

This work focuses on online dereverberation for hearing devices using the weighted prediction error (WPE) algorithm. WPE filtering requires an estimate of the target speech power spectral density (PSD). Recently deep neural networks (DNNs) have been used for this task. However, these approaches optimize the PSD estimate which only indirectly affects the WPE output, thus potentially resulting in limited dereverberation. In this paper, we propose an end-to-end approach specialized for online processing, that directly optimizes the dereverberated output signal. In addition, we propose to adapt it to the needs of different types of hearing-device users by modifying the optimization target as well as the WPE algorithm characteristics used in training. We show that the proposed end-to-end approach outperforms the traditional and conventional DNN-supported WPEs on a noise-free version of the WHAMR! dataset., Comment: \copyright 2022 IEEE. Personal use of this material is permitted. Permission from IEEE must be obtained for all other uses, in any current or future media, including reprinting/republishing this material for advertising or promotional purposes, creating new collective works, for resale or redistribution to servers or lists, or reuse of any copyrighted component of this work in other works

Recent advances in the design of neural network architectures, in particular those specialized in modeling sequences, have provided significant improvements in speech separation performance. In this work, we propose to use a bio-inspired architecture called Fully Recurrent Convolutional Neural Network (FRCNN) to solve the separation task. This model contains bottom-up, top-down and lateral connections to fuse information processed at various time-scales represented by \textit{stages}. In contrast to the traditional approach updating stages in parallel, we propose to first update the stages one by one in the bottom-up direction, then fuse information from adjacent stages simultaneously and finally fuse information from all stages to the bottom stage together. Experiments showed that this asynchronous updating scheme achieved significantly better results with much fewer parameters than the traditional synchronous updating scheme. In addition, the proposed model achieved good balance between speech separation accuracy and computational efficiency as compared to other state-of-the-art models on three benchmark datasets., Comment: Accepted by NeurIPS 2021, Demo at https://cslikai.cn/project/AFRCNN

Stochastic partial differential equations (SPDEs) are the mathematical tool of choice for modelling spatiotemporal PDE-dynamics under the influence of randomness. Based on the notion of mild solution of an SPDE, we introduce a novel neural architecture to learn solution operators of PDEs with (possibly stochastic) forcing from partially observed data. The proposed Neural SPDE model provides an extension to two popular classes of physics-inspired architectures. On the one hand, it extends Neural CDEs and variants -- continuous-time analogues of RNNs -- in that it is capable of processing incoming sequential information arriving at arbitrary spatial resolutions. On the other hand, it extends Neural Operators -- generalizations of neural networks to model mappings between spaces of functions -- in that it can parameterize solution operators of SPDEs depending simultaneously on the initial condition and a realization of the driving noise. By performing operations in the spectral domain, we show how a Neural SPDE can be evaluated in two ways, either by calling an ODE solver (emulating a spectral Galerkin scheme), or by solving a fixed point problem. Experiments on various semilinear SPDEs, including the stochastic Navier-Stokes equations, demonstrate how the Neural SPDE model is capable of learning complex spatiotemporal dynamics in a resolution-invariant way, with better accuracy and lighter training data requirements compared to alternative models, and up to 3 orders of magnitude faster than traditional solvers., Comment: Published at NeurIPS 2022

Stochastic processes are random variables with values in some space of paths. However, reducing a stochastic process to a path-valued random variable ignores its filtration, i.e. the flow of information carried by the process through time. By conditioning the process on its filtration, we introduce a family of higher order kernel mean embeddings (KMEs) that generalizes the notion of KME and captures additional information related to the filtration. We derive empirical estimators for the associated higher order maximum mean discrepancies (MMDs) and prove consistency. We then construct a filtration-sensitive kernel two-sample test able to pick up information that gets missed by the standard MMD test. In addition, leveraging our higher order MMDs we construct a family of universal kernels on stochastic processes that allows to solve real-world calibration and optimal stopping problems in quantitative finance (such as the pricing of American options) via classical kernel-based regression methods. Finally, adapting existing tests for conditional independence to the case of stochastic processes, we design a causal-discovery algorithm to recover the causal graph of structural dependencies among interacting bodies solely from observations of their multidimensional trajectories., Comment: Published at NeurIPS 2021