Autoencoders

Deep autoencoders have become a fundamental tool in various machine learning applications, ranging from dimensionality reduction and reduced order modeling of partial differential equations to anomaly detection and neural machine translation. Despite their empirical success, a solid theoretical foundation for their expressiveness remains elusive, particularly when compared to classical projection-based techniques. In this work, we aim to take a step forward in this direction by presenting a comprehensive analysis of what we refer to as symmetric autoencoders, a broad class of deep learning architectures that have surfaced frequently in the recent literature. Specifically, we introduce a formal distinction between different classes of symmetric architectures, analyzing their strengths and limitations from a mathematical perspective. For instance, we show that the reconstruction error of symmetric autoencoders with orthonormality constraints can be understood by leveraging the well-renowned Eckart-Young-Schmidt (EYS) theorem. As a byproduct of our analysis, we end up developing the EYS initialization strategy for symmetric autoencoders, which is based on an iterated application of the Singular Value Decomposition (SVD). To validate our findings, we conduct a series of numerical experiments where we benchmark our proposal against conventional deep autoencoders, discussing the importance of model design and initialization.

Reliable altitude estimation with frequency-modulated continuous wave (FMCW) radar altimeters is increasingly a challenge due to in-band interference from modern communication systems. In this paper, we present a temporal convolutional autoencoder (TCAE) that directly processes in-phase and quadrature (IQ) samples to suppress structured interference while preserving signal phase and frequency content for range estimation. The model is trained and initially evaluated within a full radar altimeter simulation chain, then further validated via over-the-air (OTA) experiments using a universal software radio peripheral (USRP)-based testbed. Results show that the TCAE reduces altitude estimation error by more than 85% compared to least mean squares (LMS) adaptive filtering under severe interference conditions, including low signal-to-interference-plus-noise ratio (SINR) and full temporal overlap between interfering and radar signals. Unlike conventional methods, the TCAE maintains phase fidelity and beat structure, enabling accurate range estimation even when interferers occupy more than one-quarter of the radar bandwidth. The implemented TCAE performs mitigation directly on fixed-length IQ windows using a single feed-forward pass and was integrated into the MATLAB/ONNX-based evaluation chain used for both simulation and OTA testing. These findings demonstrate that learned IQ-domain interference mitigation can enhance radar-altimeter resilience under a range of tested interference conditions.

The use of machine learning for modeling, understanding, and controlling large-scale physics systems is quickly gaining in popularity, with examples ranging from electromagnetism over nuclear fusion reactors and magneto-hydrodynamics to fluid mechanics and climate modeling. These systems - governed by partial differential equations - present unique challenges regarding the large number of degrees of freedom and the complex dynamics over many scales both in space and time, and additional measures to improve accuracy and sample efficiency are highly desirable. We present an end-to-end equivariant surrogate model consisting of an equivariant convolutional autoencoder and an equivariant convolutional LSTM using $G$-steerable kernels. As a case study, we consider the three-dimensional Rayleigh-B'enard convection, which describes the buoyancy-driven fluid flow between a heated bottom and a cooled top plate. While the system is E(2)-equivariant in the horizontal plane, the boundary conditions break the translational equivariance in the vertical direction. Our architecture leverages vertically stacked layers of $D_4$-steerable kernels, with additional partial kernel sharing in the vertical direction for further efficiency improvement. We demonstrate significant gains in sample and parameter efficiency, as well as a better scaling to more complex dynamics. The accompanying code is available under https://github.com/FynnFromme/equivariant-rb-forecasting.

Unsupervised anomaly detection is a challenging task. Autoencoders (AEs) or generative models are often employed to model the data distribution of normal inputs and subsequently identify anomalous, out-of-distribution inputs by high reconstruction error or low likelihood, respectively. However, AEs may generalize and achieve small reconstruction errors on abnormal inputs. We propose a decoupled training approach for anomaly detection that both an AE and a likelihood model trained with noise contrastive estimation (NCE). After training the AE, NCE estimates a probability density function, to serve as the anomaly score, on the joint space of the AE's latent representation combined with features of the reconstruction quality. To further reduce the false negative rate in NCE we systematically varying the reconstruction features to augment the training and optimize the contrastive Gaussian noise distribution. Experimental assessments on multiple benchmark datasets demonstrate that the proposed approach matches the performance of prevalent state-of-the-art anomaly detection algorithms.

Autoencoders have found widespread application in both their original deterministic form and in their variational formulation (VAEs). In scientific applications and in image processing it is often of interest to consider data that are viewed as functions; while discretisation (of differential equations arising in the sciences) or pixellation (of images) renders problems finite dimensional in practice, conceiving first of algorithms that operate on functions, and only then discretising or pixellating, leads to better algorithms that smoothly operate between resolutions. In this paper function-space versions of the autoencoder (FAE) and variational autoencoder (FVAE) are introduced, analysed, and deployed. Well-definedness of the objective governing VAEs is a subtle issue, particularly in function space, limiting applicability. For the FVAE objective to be well defined requires compatibility of the data distribution with the chosen generative model; this can be achieved, for example, when the data arise from a stochastic differential equation, but is generally restrictive. The FAE objective, on the other hand, is well defined in many situations where FVAE fails to be. Pairing the FVAE and FAE objectives with neural operator architectures that can be evaluated on any mesh enables new applications of autoencoders to inpainting, superresolution, and generative modelling of scientific data.