T. Konstantin Rusch

Wanqi Yang, Yuexiao Ma, Alexander Conzelmann et al.

Mixture-of-Experts (MoE) architectures scale model capacity through sparse expert activation, but their deployment remains memory-bound because all expert weights must reside in memory. Mixed-precision quantization can substantially reduce this footprint by assigning different bit-widths to different experts. Existing approaches, however, typically rely on calibration data to estimate expert importance and determine bit allocation. For frontier MoE LLMs, the original training data, and hence the true training distribution, is proprietary and inaccessible. As a result, calibration sets are inevitably imperfect surrogates, and this can misestimate expert utilization and lead to suboptimal bit allocation. Motivated by the substantial cross-expert quality variability observed in modern MoE models, and by the success of Heavy-Tailed Self-Regularization (HT-SR) theory at predicting neural network model quality without access to training or testing data, we propose AlphaQ, a calibration-free bit-allocation method for MoE quantization. AlphaQ draws on HT-SR theory and follows a simple principle: experts with more heavy-tailed weight spectra are typically better trained and hence should receive higher bit-widths, while experts with weaker heavy-tailed structure can be quantized more aggressively. AlphaQ operationalizes this principle by measuring expert-wise spectral heavy-tailedness and solving a budget-constrained optimization problem that minimizes total quantization error under a global bit-budget constraint. Across several MoE models, AlphaQ consistently outperforms calibration-based baselines under matched bit budgets. Notably, on Qwen1.5-MoE, AlphaQ achieves near full-precision accuracy with an average expert precision of only 3.5 bits, while delivering more than 4$\times$ memory compression. Our code is available at https://github.com/Superone77/AlphaQ.

T. Konstantin Rusch, Michael M. Bronstein, Siddhartha Mishra · eth-zurich

Node features of graph neural networks (GNNs) tend to become more similar with the increase of the network depth. This effect is known as over-smoothing, which we axiomatically define as the exponential convergence of suitable similarity measures on the node features. Our definition unifies previous approaches and gives rise to new quantitative measures of over-smoothing. Moreover, we empirically demonstrate this behavior for several over-smoothing measures on different graphs (small-, medium-, and large-scale). We also review several approaches for mitigating over-smoothing and empirically test their effectiveness on real-world graph datasets. Through illustrative examples, we demonstrate that mitigating over-smoothing is a necessary but not sufficient condition for building deep GNNs that are expressive on a wide range of graph learning tasks. Finally, we extend our definition of over-smoothing to the rapidly emerging field of continuous-time GNNs.

T. Konstantin Rusch, Benjamin P. Chamberlain, Michael W. Mahoney et al. · eth-zurich

We present Gradient Gating (G$^2$), a novel framework for improving the performance of Graph Neural Networks (GNNs). Our framework is based on gating the output of GNN layers with a mechanism for multi-rate flow of message passing information across nodes of the underlying graph. Local gradients are harnessed to further modulate message passing updates. Our framework flexibly allows one to use any basic GNN layer as a wrapper around which the multi-rate gradient gating mechanism is built. We rigorously prove that G$^2$ alleviates the oversmoothing problem and allows the design of deep GNNs. Empirical results are presented to demonstrate that the proposed framework achieves state-of-the-art performance on a variety of graph learning tasks, including on large-scale heterophilic graphs.

Francesco Di Giovanni, T. Konstantin Rusch, Michael M. Bronstein et al. · eth-zurich

Graph Neural Networks (GNNs) are the state-of-the-art model for machine learning on graph-structured data. The most popular class of GNNs operate by exchanging information between adjacent nodes, and are known as Message Passing Neural Networks (MPNNs). Given their widespread use, understanding the expressive power of MPNNs is a key question. However, existing results typically consider settings with uninformative node features. In this paper, we provide a rigorous analysis to determine which function classes of node features can be learned by an MPNN of a given capacity. We do so by measuring the level of pairwise interactions between nodes that MPNNs allow for. This measure provides a novel quantitative characterization of the so-called over-squashing effect, which is observed to occur when a large volume of messages is aggregated into fixed-size vectors. Using our measure, we prove that, to guarantee sufficient communication between pairs of nodes, the capacity of the MPNN must be large enough, depending on properties of the input graph structure, such as commute times. For many relevant scenarios, our analysis results in impossibility statements in practice, showing that over-squashing hinders the expressive power of MPNNs. We validate our theoretical findings through extensive controlled experiments and ablation studies.

Francesco M. Ruscio, T. Konstantin Rusch

Flow Matching typically relies on white noise sources, a choice often misaligned with the power spectra of natural data, which tend to decay with frequency. To address this, we introduce Low-Pass Flow Matching, a variant of Flow Matching based on an operator-modulated interpolant. This formulation induces a time-varying spectral bias that transitions from the source spectrum to a frequency-decaying bias as the path approaches the data. We validate our method on unconditional image generation tasks, including the scientific Galaxy10 dataset. Empirically, we show that our method is particularly effective when paired with adaptive ODE solvers, where it improves or preserves sample quality while substantially reducing sampling cost compared to standard baselines.

Léonard Equer, T. Konstantin Rusch, Siddhartha Mishra · eth-zurich

We propose a novel multi-scale message passing neural network algorithm for learning the solutions of time-dependent PDEs. Our algorithm possesses both temporal and spatial multi-scale resolution features by incorporating multi-scale sequence models and graph gating modules in the encoder and processor, respectively. Benchmark numerical experiments are presented to demonstrate that the proposed algorithm outperforms baselines, particularly on a PDE with a range of spatial and temporal scales.

Philipp Nazari, T. Konstantin Rusch

Linear attention offers a computationally efficient yet expressive alternative to softmax attention. However, recent empirical results indicate that the state of trained linear attention models often exhibits a low-rank structure, suggesting that these models underexploit their capacity in practice. To illuminate this phenomenon, we provide a theoretical analysis of the role of rank in linear attention, revealing that low effective rank can affect retrieval error by amplifying query noise. In addition to these theoretical insights, we conjecture that the low-rank states can be substantially reduced post-training with only minimal performance degradation, yielding faster and more memory-efficient models. To this end, we propose a novel hardware-aware approach that structurally prunes key and query matrices, reducing the state size while retaining compatibility with existing CUDA kernels. We adapt several existing pruning strategies to fit our framework and, building on our theoretical analysis, propose a novel structured pruning method based on a rank-revealing QR decomposition. Our empirical results, evaluated across models of varying sizes and on various downstream tasks, demonstrate the effectiveness of our state reduction framework. We highlight that our framework enables the removal of 50% of the query and key channels at only a marginal increase in perplexity. The code for this project can be found at https://github.com/camail-official/LinearAttentionPruning.

T. Konstantin Rusch, Nathan Kirk, Michael M. Bronstein et al. · eth-zurich

Discrepancy is a well-known measure for the irregularity of the distribution of a point set. Point sets with small discrepancy are called low-discrepancy and are known to efficiently fill the space in a uniform manner. Low-discrepancy points play a central role in many problems in science and engineering, including numerical integration, computer vision, machine perception, computer graphics, machine learning, and simulation. In this work, we present the first machine learning approach to generate a new class of low-discrepancy point sets named Message-Passing Monte Carlo (MPMC) points. Motivated by the geometric nature of generating low-discrepancy point sets, we leverage tools from Geometric Deep Learning and base our model on Graph Neural Networks. We further provide an extension of our framework to higher dimensions, which flexibly allows the generation of custom-made points that emphasize the uniformity in specific dimensions that are primarily important for the particular problem at hand. Finally, we demonstrate that our proposed model achieves state-of-the-art performance superior to previous methods by a significant margin. In fact, MPMC points are empirically shown to be either optimal or near-optimal with respect to the discrepancy for low dimension and small number of points, i.e., for which the optimal discrepancy can be determined. Code for generating MPMC points can be found at https://github.com/tk-rusch/MPMC.

Michael Etienne Van Huffel, Nathan Kirk, Makram Chahine et al. · eth-zurich

Low-discrepancy points are designed to efficiently fill the space in a uniform manner. This uniformity is highly advantageous in many problems in science and engineering, including in numerical integration, computer vision, machine perception, computer graphics, machine learning, and simulation. Whereas most previous low-discrepancy constructions rely on abstract algebra and number theory, Message-Passing Monte Carlo (MPMC) was recently introduced to exploit machine learning methods for generating point sets with lower discrepancy than previously possible. However, MPMC is limited to generating point sets and cannot be extended to low-discrepancy sequences (LDS), i.e., sequences of points in which every prefix has low discrepancy, a property essential for many applications. To address this limitation, we introduce Neural Low-Discrepancy Sequences ($NeuroLDS$), the first machine learning-based framework for generating LDS. Drawing inspiration from classical LDS, we train a neural network to map indices to points such that the resulting sequences exhibit minimal discrepancy across all prefixes. To this end, we deploy a two-stage learning process: supervised approximation of classical constructions followed by unsupervised fine-tuning to minimize prefix discrepancies. We demonstrate that $NeuroLDS$ outperforms all previous LDS constructions by a significant margin with respect to discrepancy measures. Moreover, we demonstrate the effectiveness of $NeuroLDS$ across diverse applications, including numerical integration, robot motion planning, and scientific machine learning. These results highlight the promise and broad significance of Neural Low-Discrepancy Sequences. Our code can be found at https://github.com/camail-official/neuro-lds.

Makram Chahine, Philipp Nazari, Daniela Rus et al. · eth-zurich

State Space Models (SSMs), developed to tackle long sequence modeling tasks efficiently, offer both parallelizable training and fast inference. At their core are recurrent dynamical systems that maintain a hidden state, with update costs scaling with the state dimension. A key design challenge is striking the right balance between maximizing expressivity and limiting this computational burden. Control theory, and more specifically Hankel singular value analysis, provides a potent framework for the measure of energy for each state, as well as the balanced truncation of the original system down to a smaller representation with performance guarantees. Leveraging the eigenvalue stability properties of Hankel matrices, we apply this lens to SSMs \emph{during training}, where only dimensions of high influence are identified and preserved. Our approach, \textsc{CompreSSM}, applies to Linear Time-Invariant SSMs such as Linear Recurrent Units, but is also extendable to selective models. Experiments show that in-training reduction significantly accelerates optimization while preserving expressivity, with compressed models retaining task-critical structure lost by models trained directly at smaller dimension. In other words, SSMs that begin large and shrink during training achieve computational efficiency while maintaining higher performance. Project code is available at github.com/camail-official/compressm.

Ahmed A. Elhag, T. Konstantin Rusch, Francesco Di Giovanni et al. · eth-zurich

Incorporating equivariance as an inductive bias into deep learning architectures to take advantage of the data symmetry has been successful in multiple applications, such as chemistry and dynamical systems. In particular, roto-translations are crucial for effectively modeling geometric graphs and molecules, where understanding the 3D structures enhances generalization. However, equivariant models often pose challenges due to their high computational complexity. In this paper, we introduce REMUL, a training procedure for approximating equivariance with multitask learning. We show that unconstrained models (which do not build equivariance into the architecture) can learn approximate symmetries by minimizing an additional simple equivariance loss. By formulating equivariance as a new learning objective, we can control the level of approximate equivariance in the model. Our method achieves competitive performance compared to equivariant baselines while being $10 \times$ faster at inference and $2.5 \times$ at training.

Nathan Kirk, T. Konstantin Rusch, Jakob Zech et al. · eth-zurich

Message-Passing Monte Carlo (MPMC) was recently introduced as a novel low-discrepancy sampling approach leveraging tools from geometric deep learning. While originally designed for generating uniform point sets, we extend this framework to sample from general multivariate probability distributions with known probability density function. Our proposed method, Stein-Message-Passing Monte Carlo (Stein-MPMC), minimizes a kernelized Stein discrepancy, ensuring improved sample quality. Finally, we show that Stein-MPMC outperforms competing methods, such as Stein Variational Gradient Descent and (greedy) Stein Points, by achieving a lower Stein discrepancy.

Jared Boyer, T. Konstantin Rusch, Daniela Rus · eth-zurich

State-space models (SSMs) are a class of networks for sequence learning that benefit from fixed state size and linear complexity with respect to sequence length, contrasting the quadratic scaling of typical attention mechanisms. Inspired from observations in neuroscience, Linear Oscillatory State-Space models (LinOSS) are a recently proposed class of SSMs constructed from layers of discretized forced harmonic oscillators. Although these models perform competitively, leveraging fast parallel scans over diagonal recurrent matrices and achieving state-of-the-art performance on tasks with sequence length up to 50k, LinOSS models rely on rigid energy dissipation ("forgetting") mechanisms that are inherently coupled to the time scale of state evolution. As forgetting is a crucial mechanism for long-range reasoning, we demonstrate the representational limitations of these models and introduce Damped Linear Oscillatory State-Space models (D-LinOSS), a more general class of oscillatory SSMs that learn to dissipate latent state energy on arbitrary time scales. We analyze the spectral distribution of the model's recurrent matrices and prove that the SSM layers exhibit stable dynamics under a simple, flexible parameterization. Without introducing additional complexity, D-LinOSS consistently outperforms previous LinOSS methods on long-range learning tasks, achieves faster convergence, and reduces the hyperparameter search space by 50%.

Rodney Lafuente-Mercado, Daniela Rus, T. Konstantin Rusch

How much does a trained RL policy actually use its past observations? We propose \emph{Temporal Range}, a model-agnostic metric that treats first-order sensitivities of multiple vector outputs across a temporal window to the input sequence as a temporal influence profile and summarizes it by the magnitude-weighted average lag. Temporal Range is computed via reverse-mode automatic differentiation from the Jacobian blocks $\partial y_s/\partial x_t\in\mathbb{R}^{c\times d}$ averaged over final timesteps $s\in\{t+1,\dots,T\}$ and is well-characterized in the linear setting by a small set of natural axioms. Across diagnostic and control tasks (POPGym; flicker/occlusion; Copy-$k$) and architectures (MLPs, RNNs, SSMs), Temporal Range (i) remains small in fully observed control, (ii) scales with the task's ground-truth lag in Copy-$k$, and (iii) aligns with the minimum history window required for near-optimal return as confirmed by window ablations. We also report Temporal Range for a compact Long Expressive Memory (LEM) policy trained on the task, using it as a proxy readout of task-level memory. Our axiomatic treatment draws on recent work on range measures, specialized here to temporal lag and extended to vector-valued outputs in the RL setting. Temporal Range thus offers a practical per-sequence readout of memory dependence for comparing agents and environments and for selecting the shortest sufficient context.

Maximilian Stölzle, T. Konstantin Rusch, Zach J. Patterson et al. · eth-zurich

Learning from demonstration provides a sample-efficient approach to acquiring complex behaviors, enabling robots to move robustly, compliantly, and with fluidity. In this context, Dynamic Motion Primitives offer built - in stability and robustness to disturbances but often struggle to capture complex periodic behaviors. Moreover, they are limited in their ability to interpolate between different tasks. These shortcomings substantially narrow their applicability, excluding a wide class of practically meaningful tasks such as locomotion and rhythmic tool use. In this work, we introduce Orbitally Stable Motion Primitives (OSMPs) - a framework that combines a learned diffeomorphic encoder with a supercritical Hopf bifurcation in latent space, enabling the accurate acquisition of periodic motions from demonstrations while ensuring formal guarantees of orbital stability and transverse contraction. Furthermore, by conditioning the bijective encoder on the task, we enable a single learned policy to represent multiple motion objectives, yielding consistent zero-shot generalization to unseen motion objectives within the training distribution. We validate the proposed approach through extensive simulation and real-world experiments across a diverse range of robotic platforms - from collaborative arms and soft manipulators to a bio-inspired rigid-soft turtle robot - demonstrating its versatility and effectiveness in consistently outperforming state-of-the-art baselines such as diffusion policies, among others.

Samuel Lanthaler, T. Konstantin Rusch, Siddhartha Mishra

Coupled oscillators are being increasingly used as the basis of machine learning (ML) architectures, for instance in sequence modeling, graph representation learning and in physical neural networks that are used in analog ML devices. We introduce an abstract class of neural oscillators that encompasses these architectures and prove that neural oscillators are universal, i.e, they can approximate any continuous and casual operator mapping between time-varying functions, to desired accuracy. This universality result provides theoretical justification for the use of oscillator based ML systems. The proof builds on a fundamental result of independent interest, which shows that a combination of forced harmonic oscillators with a nonlinear read-out suffices to approximate the underlying operators.

T. Konstantin Rusch, Benjamin P. Chamberlain, James Rowbottom et al.

We propose Graph-Coupled Oscillator Networks (GraphCON), a novel framework for deep learning on graphs. It is based on discretizations of a second-order system of ordinary differential equations (ODEs), which model a network of nonlinear controlled and damped oscillators, coupled via the adjacency structure of the underlying graph. The flexibility of our framework permits any basic GNN layer (e.g. convolutional or attentional) as the coupling function, from which a multi-layer deep neural network is built up via the dynamics of the proposed ODEs. We relate the oversmoothing problem, commonly encountered in GNNs, to the stability of steady states of the underlying ODE and show that zero-Dirichlet energy steady states are not stable for our proposed ODEs. This demonstrates that the proposed framework mitigates the oversmoothing problem. Moreover, we prove that GraphCON mitigates the exploding and vanishing gradients problem to facilitate training of deep multi-layer GNNs. Finally, we show that our approach offers competitive performance with respect to the state-of-the-art on a variety of graph-based learning tasks.

T. Konstantin Rusch, Siddhartha Mishra, N. Benjamin Erichson et al.

We propose a novel method called Long Expressive Memory (LEM) for learning long-term sequential dependencies. LEM is gradient-based, it can efficiently process sequential tasks with very long-term dependencies, and it is sufficiently expressive to be able to learn complicated input-output maps. To derive LEM, we consider a system of multiscale ordinary differential equations, as well as a suitable time-discretization of this system. For LEM, we derive rigorous bounds to show the mitigation of the exploding and vanishing gradients problem, a well-known challenge for gradient-based recurrent sequential learning methods. We also prove that LEM can approximate a large class of dynamical systems to high accuracy. Our empirical results, ranging from image and time-series classification through dynamical systems prediction to speech recognition and language modeling, demonstrate that LEM outperforms state-of-the-art recurrent neural networks, gated recurrent units, and long short-term memory models.

T. Konstantin Rusch, Siddhartha Mishra

The design of recurrent neural networks (RNNs) to accurately process sequential inputs with long-time dependencies is very challenging on account of the exploding and vanishing gradient problem. To overcome this, we propose a novel RNN architecture which is based on a structure preserving discretization of a Hamiltonian system of second-order ordinary differential equations that models networks of oscillators. The resulting RNN is fast, invertible (in time), memory efficient and we derive rigorous bounds on the hidden state gradients to prove the mitigation of the exploding and vanishing gradient problem. A suite of experiments are presented to demonstrate that the proposed RNN provides state of the art performance on a variety of learning tasks with (very) long-time dependencies.

T. Konstantin Rusch, Siddhartha Mishra

Circuits of biological neurons, such as in the functional parts of the brain can be modeled as networks of coupled oscillators. Inspired by the ability of these systems to express a rich set of outputs while keeping (gradients of) state variables bounded, we propose a novel architecture for recurrent neural networks. Our proposed RNN is based on a time-discretization of a system of second-order ordinary differential equations, modeling networks of controlled nonlinear oscillators. We prove precise bounds on the gradients of the hidden states, leading to the mitigation of the exploding and vanishing gradient problem for this RNN. Experiments show that the proposed RNN is comparable in performance to the state of the art on a variety of benchmarks, demonstrating the potential of this architecture to provide stable and accurate RNNs for processing complex sequential data.

Siddhartha Mishra, T. Konstantin Rusch

We propose a deep supervised learning algorithm based on low-discrepancy sequences as the training set. By a combination of theoretical arguments and extensive numerical experiments we demonstrate that the proposed algorithm significantly outperforms standard deep learning algorithms that are based on randomly chosen training data, for problems in moderately high dimensions. The proposed algorithm provides an efficient method for building inexpensive surrogates for many underlying maps in the context of scientific computing.