Once closed-form expressions for these Jacobians are derived, it remains to substitute those expressions into (16). The following identity (often termed the "vec" rule) will To depict the spatial topographies of the latent components measured on the EEG and fMRI analyses, the "forward-model" [ The results of the comparison are shown in Fig S1, where it is clear that the signal fidelity of the GCs (right panel) significantly exceeds those yielded by PCA (left) and ICA (middle). GCA is only able to recover sources with temporal dependencies (i.e., s Both the single electrodes and Granger components exhibit two pronounced peaks in the spectra: one near 2 Hz ("delta" Fig S3 shows the corresponding result for the left motor imagery condition. EEG motor imagery dataset described in the main text. For each technique, the first 6 components are presented.

Recently developed applications in the field of machine learning and computational physics rely on automatic differentiation techniques, that require stable and efficient linear algebra gradient computations. This technical note provides a comprehensive and detailed discussion of the derivative of the truncated singular and eigenvalue decomposition. It summarizes previous work and builds on them with an extensive description of how to derive the relevant terms. A main focus is correctly expressing the derivative in terms of the truncated part, despite lacking knowledge of the full decomposition.

We sincerely thank all the reviewers for their meticulous work and their helpful and detailed comments. We will gladly incorporate them! Thank you for this suggestion. For the Stable manifold theorem in line 30, we refer to Chapter 5 Theorem III.7 in "Global Stability of Dynamical Systems" by M. Shub and in line 130, we refer to Section 2.7 in "Differential Equations and Dynamical Systems" by L. "Might be interesting to add a note... Is this correct?" "Does the paper... stuck in a saddle point" The last parts of the statements of Theorem 4.1 and Corollary 4.2 assert "This is of course... that area of study."

We argue that this objective is insufficient to ensure doctors'

We present a conditional diffusion model for electromagnetic inverse design that generates structured media geometries directly from target differential scattering cross-section profiles, bypassing expensive iterative optimization. Our 1D U-Net architecture with Feature-wise Linear Modulation learns to map desired angular scattering patterns to 2x2 dielectric sphere structure, naturally handling the non-uniqueness of inverse problems by sampling diverse valid designs. Trained on 11,000 simulated metasurfaces, the model achieves median MPE below 19% on unseen targets (best: 1.39%), outperforming CMA-ES evolutionary optimization while reducing design time from hours to seconds. These results demonstrate that employing diffusion models is promising for advancing electromagnetic inverse design research, potentially enabling rapid exploration of complex metasurface architectures and accelerating the development of next-generation photonic and wireless communication systems. The code is publicly available at https://github.com/mikzuker/inverse_design_metasurface_generation.

Consistency models are a class of generative models that enable few-step generation for diffusion and flow matching models. While consistency models have achieved promising results on Euclidean domains like images, their applications to Riemannian manifolds remain challenging due to the curved geometry. In this work, we propose the Riemannian Consistency Model (RCM), which, for the first time, enables few-step consistency modeling while respecting the intrinsic manifold constraint imposed by the Riemannian geometry. Leveraging the covariant derivative and exponential-map-based parameterization, we derive the closed-form solutions for both discrete- and continuous-time training objectives for RCM. We then demonstrate theoretical equivalence between the two variants of RCM: Riemannian consistency distillation (RCD) that relies on a teacher model to approximate the marginal vector field, and Riemannian consistency training (RCT) that utilizes the conditional vector field for training. We further propose a simplified training objective that eliminates the need for the complicated differential calculation. Finally, we provide a unique kinematics perspective for interpreting the RCM objective, offering new theoretical angles. Through extensive experiments, we manifest the superior generative quality of RCM in few-step generation on various non-Euclidean manifolds, including flat-tori, spheres, and the 3D rotation group SO(3).

Once closed-form expressions for these Jacobians are derived, it remains to substitute those expressions into (16). The following identity (often termed the "vec" rule) will To depict the spatial topographies of the latent components measured on the EEG and fMRI analyses, the "forward-model" [ The results of the comparison are shown in Fig S1, where it is clear that the signal fidelity of the GCs (right panel) significantly exceeds those yielded by PCA (left) and ICA (middle). GCA is only able to recover sources with temporal dependencies (i.e., s Both the single electrodes and Granger components exhibit two pronounced peaks in the spectra: one near 2 Hz ("delta" Fig S3 shows the corresponding result for the left motor imagery condition. EEG motor imagery dataset described in the main text. For each technique, the first 6 components are presented.

We sincerely thank all the reviewers for their meticulous work and their helpful and detailed comments. We will gladly incorporate them! Thank you for this suggestion. For the Stable manifold theorem in line 30, we refer to Chapter 5 Theorem III.7 in "Global Stability of Dynamical Systems" by M. Shub and in line 130, we refer to Section 2.7 in "Differential Equations and Dynamical Systems" by L. "Might be interesting to add a note... Is this correct?" "Does the paper... stuck in a saddle point" The last parts of the statements of Theorem 4.1 and Corollary 4.2 assert "This is of course... that area of study."

Queen bee presence is essential for the health and stability of honeybee colonies, yet current monitoring methods rely on manual inspections that are labor-intensive, disruptive, and impractical for large-scale beekeeping. While recent audio-based approaches have shown promise, they often require high power consumption, complex preprocessing, and are susceptible to ambient noise. To overcome these limitations, we propose a lightweight, multimodal system for queen detection based on environmental sensor fusion-specifically, temperature, humidity, and pressure differentials between the inside and outside of the hive. Our approach employs quantized decision tree inference on a commercial STM32 microcontroller, enabling real-time, low-power edge computing without compromising accuracy. We show that our system achieves over 99% queen detection accuracy using only environmental inputs, with audio features offering no significant performance gain. This work presents a scalable and sustainable solution for non-invasive hive monitoring, paving the way for autonomous, precision beekeeping using off-the-shelf, energy-efficient hardware.

The Expectation-Maximisation (EM) algorithm is a central tool in statistics and machine learning, widely used for latent-variable models such as Gaussian Mixture Models (GMMs). Despite its ubiquity, EM is typically treated as a non-differentiable black box, preventing its integration into modern learning pipelines where end-to-end gradient propagation is essential. In this work, we present and compare several differentiation strategies for EM, from full automatic differentiation to approximate methods, assessing their accuracy and computational efficiency. As a key application, we leverage this differentiable EM in the computation of the Mixture Wasserstein distance $\mathrm{MW}_2$ between GMMs, allowing $\mathrm{MW}_2$ to be used as a differentiable loss in imaging and machine learning tasks. To complement our practical use of $\mathrm{MW}_2$, we contribute a novel stability result which provides theoretical justification for the use of $\mathrm{MW}_2$ with EM, and also introduce a novel unbalanced variant of $\mathrm{MW}_2$. Numerical experiments on barycentre computation, colour and style transfer, image generation, and texture synthesis illustrate the versatility and effectiveness of the proposed approach in different settings.