Daily Papers

In this article, we study the consistency of the template estimation with the Fr\'echet mean in quotient spaces. The Fr\'echet mean in quotient spaces is often used when the observations are deformed or transformed by a group action. We show that in most cases this estimator is actually inconsistent. We exhibit a sufficient condition for this inconsistency, which amounts to the folding of the distribution of the noisy template when it is projected to the quotient space. This condition appears to be fulfilled as soon as the support of the noise is large enough. To quantify this inconsistency we provide lower and upper bounds of the bias as a function of the variability (the noise level). This shows that the consistency bias cannot be neglected when the variability increases.

To characterize the location (mean, median) of a set of graphs, one needs a notion of centrality that is adapted to metric spaces, since graph sets are not Euclidean spaces. A standard approach is to consider the Frechet mean. In this work, we equip a set of graphs with the pseudometric defined by the norm between the eigenvalues of their respective adjacency matrix. Unlike the edit distance, this pseudometric reveals structural changes at multiple scales, and is well adapted to studying various statistical problems for graph-valued data. We describe an algorithm to compute an approximation to the sample Frechet mean of a set of undirected unweighted graphs with a fixed size using this pseudometric.

Understanding the geometric structure of pre-trained language model embeddings matters for interpretability and safety. We ask whether sentence-level classification signal lives in the Riemannian geometry of contextual token embeddings, and probe it by extracting per-token pullback metrics from a learned encoder's analytical Jacobian and aggregating them with the Fréchet mean on the symmetric positive definite (SPD) manifold; we call this procedure Riemannian Mean Pooling (RMP). Across three datasets with non-trivial linguistic structure (CoLA, CREAK, RTE), RMP outperforms Euclidean mean pooling, while on FEVER-Symmetric, a benchmark constructed to remove annotation-driven lexical artifacts, the method correctly stays at chance. Ablations show that a randomly initialised encoder combined with Fréchet aggregation already beats Euclidean pooling on two of the three signal-bearing datasets, localising the source of the gain to the geometric aggregation rather than to learned manifold structure; the trained encoder contributes additional signal specifically on CREAK, the most knowledge-heavy of the three signal-bearing datasets.

Model merging offers a promising avenue for knowledge integration and parallel development without retraining. Yet, existing methods either ignore the geometry of the loss landscape or rely on intractable full-space Hessian approximations. We propose EpiMer, a framework that casts model merging as solving the Fréchet mean on a Riemannian manifold and restricts the computation to a low-rank subspace spanned by the task vectors. With the expected Hessian as the metric, we reveal a connection between local curvature and epistemic uncertainty of the parameters. Our theoretical analysis decomposes the merging error bound into the subspace Fréchet variance and the residual energy, and provides a closed-form characterization of when curvature-aware merging provably outperforms flat-geometry methods. In addition, our framework unifies both curvature-aware methods and recent spectral methods as special cases of the subspace Fréchet mean with different geometric metrics. Merging fine-tuned CLIP-ViT models on eight image classification tasks, Epistemic Merging strictly outperforms the baselines on all three CLIP-ViT backbones at matched rank, improving the across-task average accuracy and worst-task accuracy on every backbone.

Machine learning models -- including prominent zero-shot models -- are often trained on datasets whose labels are only a small proportion of a larger label space. Such spaces are commonly equipped with a metric that relates the labels via distances between them. We propose a simple approach to exploit this information to adapt the trained model to reliably predict new classes -- or, in the case of zero-shot prediction, to improve its performance -- without any additional training. Our technique is a drop-in replacement of the standard prediction rule, swapping argmax with the Fr\'echet mean. We provide a comprehensive theoretical analysis for this approach, studying (i) learning-theoretic results trading off label space diameter, sample complexity, and model dimension, (ii) characterizations of the full range of scenarios in which it is possible to predict any unobserved class, and (iii) an optimal active learning-like next class selection procedure to obtain optimal training classes for when it is not possible to predict the entire range of unobserved classes. Empirically, using easily-available external metrics, our proposed approach, Loki, gains up to 29.7% relative improvement over SimCLR on ImageNet and scales to hundreds of thousands of classes. When no such metric is available, Loki can use self-derived metrics from class embeddings and obtains a 10.5% improvement on pretrained zero-shot models such as CLIP.

In statistics, independent, identically distributed random samples do not carry a natural ordering, and their statistics are typically invariant with respect to permutations of their order. Thus, an n-sample in a space M can be considered as an element of the quotient space of M^n modulo the permutation group. The present paper takes this definition of sample space and the related concept of orbit types as a starting point for developing a geometric perspective on statistics. We aim at deriving a general mathematical setting for studying the behavior of empirical and population means in spaces ranging from smooth Riemannian manifolds to general stratified spaces. We fully describe the orbifold and path-metric structure of the sample space when M is a manifold or path-metric space, respectively. These results are non-trivial even when M is Euclidean. We show that the infinite sample space exists in a Gromov-Hausdorff type sense and coincides with the Wasserstein space of probability distributions on M. We exhibit Fr\'echet means and k-means as metric projections onto 1-skeleta or k-skeleta in Wasserstein space, and we define a new and more general notion of polymeans. This geometric characterization via metric projections applies equally to sample and population means, and we use it to establish asymptotic properties of polymeans such as consistency and asymptotic normality.

While recent few-step sampling text-to-audio generation models like MeanAudio substantially accelerate generation by modeling average velocities, their strict one-step generation quality still lags significantly behind multi-step counterparts. We propose FdAudio to bridge this gap. Unlike MeanAudio, which relies solely on regression against target velocity fields, our post-training approach optimizes the final one-step distribution directly across pre-trained embedding spaces via a multi-representation Fréchet-distance (FD) loss. Crucially, to prevent the multi-step degradation that naive post-training with FD-loss causes, we introduce a MeanFlow consistency objective as a structural anchor. Results demonstrate that FdAudio establishes state-of-the-art one-step T2A generation quality among few-step systems, yielding an 11.4% reduction in FD score and a 28.8% improvement in FAD score relative to the baseline MeanAudio framework. Notably, we solve FD post-training's naive multi-step degradation issue by proposing the MeanFlow anchor, enabling a 25-step sampling path to maintain high-fidelity audio synthesis that matches or surpasses strong multi-step models at a fraction of their computational latency.

Few-step diffusion and flow-matching text-to-speech (TTS) models are usually trained with local objectives, such as conditional flow matching, reconstruction, and stop prediction. These losses provide stable optimization, but they never ask whether sampled speech follows the distribution of high-quality speech. We propose Speech Representation Fr'echet Distance loss (SR-FD), a training-time distributional regularizer for tokenizer-free flow-matching autoregressive TTS. During fine-tuning, the model synthesizes speech with the same few-step sampler used at deployment, and SR-FD matches the mean and covariance of frozen Whisper and CTC features of this speech to reference statistics computed offline from three complementary content targets. The loss requires no discriminator and no inference-time computation. On Seed-TTS English, four-step SR-FD fine-tuning reduces WER from the original four-step VoxCPM2 baseline's 2.2279% to 1.4147%, a 36.5% relative reduction, and also surpasses the original ten-step baseline at 1.7366%; both gains are significant under an utterance-level paired bootstrap. Speaker similarity and objective quality proxies are preserved at the ten-step level, and an error analysis shows the gain comes from content substitutions across all prompt lengths. SR-FD is thus an intelligibility-improving distributional regularizer for few-step TTS.

Determining whether two sets of images belong to the same or different distributions or domains is a crucial task in modern medical image analysis and deep learning; for example, to evaluate the output quality of image generative models. Currently, metrics used for this task either rely on the (potentially biased) choice of some downstream task, such as segmentation, or adopt task-independent perceptual metrics (e.g., Fréchet Inception Distance/FID) from natural imaging, which we show insufficiently capture anatomical features. To this end, we introduce a new perceptual metric tailored for medical images, FRD (Fréchet Radiomic Distance), which utilizes standardized, clinically meaningful, and interpretable image features. We show that FRD is superior to other image distribution metrics for a range of medical imaging applications, including out-of-domain (OOD) detection, the evaluation of image-to-image translation (by correlating more with downstream task performance as well as anatomical consistency and realism), and the evaluation of unconditional image generation. Moreover, FRD offers additional benefits such as stability and computational efficiency at low sample sizes, sensitivity to image corruptions and adversarial attacks, feature interpretability, and correlation with radiologist-perceived image quality. Additionally, we address key gaps in the literature by presenting an extensive framework for the multifaceted evaluation of image similarity metrics in medical imaging -- including the first large-scale comparative study of generative models for medical image translation -- and release an accessible codebase to facilitate future research. Our results are supported by thorough experiments spanning a variety of datasets, modalities, and downstream tasks, highlighting the broad potential of FRD for medical image analysis.

Standard methods for detecting discontinuities in conditional means are not applicable to outcomes that are complex, non-Euclidean objects like distributions, networks, or covariance matrices. This article develops a nonparametric test for jumps in conditional means when outcomes lie in a non-Euclidean metric space. Using local Fr\'echet regressionx2014which generalizes standard regression to metric-space valued datax2014the method estimates a mean path on either side of a candidate cutoff, extending existing k-sample tests to a flexible regression setting. Key theoretical contributions include a central limit theorem for the local estimator of the conditional Fr\'echet variance and the asymptotic validity and consistency of the proposed test. Simulations confirm nominal size control and robust power in finite samples. Two applications demonstrate the method's value by revealing effects invisible to scalar-based tests. First, I detect a sharp change in work-from-home compositions at Washington State's income threshold for non-compete enforceability during COVID-19, highlighting remote work's role as a bargaining margin. Second, I find that countries restructure their input-output networks after losing preferential US trade access. These findings underscore that analyzing regression functions within their native metric spaces can reveal structural discontinuities that scalar summaries would miss.

Model merging aims to combine multiple models into one without additional training. Naïve parameter-space averaging can be fragile under architectural symmetries, as their geometry does not take them into account. In this work we show that not only the geometry, but also the averaging procedure itself, must be symmetry-invariant to achieve symmetry-aware merges. Consequently, we propose a general solution: merging as Fréchet averaging, i.e., selecting parameters that minimize a sum of geodesic distances on an appropriate manifold. In this view, the key design choice is the overall geometry, i.e., the choice of metric, manifold, and distance approximation, that determines what it means for two models to be "close". We show that Fréchet averaging, combined with simplifying assumptions, contains Fisher merging. Building on this, we examine the particular case of low-rank adapters (LoRA), whose symmetries induce a distinct geometry: that of a quotient manifold. We outline the limitations of current LoRA merging methods, propose a practical algorithm for this setting, and show how they compare with other commonly used approaches.

Feature embeddings acquired from pretrained models are widely used in medical applications of deep learning to assess the characteristics of datasets; e.g. to determine the quality of synthetic, generated medical images. The Fréchet Inception Distance (FID) is one popular synthetic image quality metric that relies on the assumption that the characteristic features of the data can be detected and encoded by an InceptionV3 model pretrained on ImageNet1K (natural images). While it is widely known that this makes it less effective for applications involving medical images, the extent to which the metric fails to capture meaningful differences in image characteristics is not obviously known. Here, we use Monte Carlo dropout to compute the predictive variance in the FID as well as a supplemental estimate of the predictive variance in the feature embedding model's latent representations. We show that the magnitudes of the predictive variances considered exhibit varying degrees of correlation with the extent to which test inputs (ImageNet1K validation set augmented at various strengths, and other external datasets) are out-of-distribution relative to its training data, providing some insight into the effectiveness of their use as indicators of the trustworthiness of the FID.

In the realm of image synthesis, achieving fidelity to a reference image while adhering to conditional prompts remains a significant challenge. This paper proposes a novel approach that integrates a diffusion model with latent space manipulation and gradient-based selective attention mechanisms to address this issue. Leveraging Grad-SAM (Gradient-based Selective Attention Manipulation), we analyze the cross attention maps of the cross attention layers and gradients for the denoised latent vector, deriving importance scores of elements of denoised latent vector related to the subject of interest. Using this information, we create masks at specific timesteps during denoising to preserve subjects while seamlessly integrating the reference image features. This approach ensures the faithful formation of subjects based on conditional prompts, while concurrently refining the background for a more coherent composition. Our experiments on places365 dataset demonstrate promising results, with our proposed model achieving the lowest mean and median Frechet Inception Distance (FID) scores compared to baseline models, indicating superior fidelity preservation. Furthermore, our model exhibits competitive performance in aligning the generated images with provided textual descriptions, as evidenced by high CLIP scores. These results highlight the effectiveness of our approach in both fidelity preservation and textual context preservation, offering a significant advancement in text-to-image synthesis tasks.

Efficient video transmission is essential for seamless communication and collaboration within the visually-driven digital landscape. To achieve low latency and high-quality video transmission over a bandwidth-constrained noisy wireless channel, we propose a stable diffusion (SD)-based goal-oriented semantic communication (GSC) framework. In this framework, we first design a semantic encoder that effectively identify the keyframes from video and extract the relevant semantic information (SI) to reduce the transmission data size. We then develop a semantic decoder to reconstruct the keyframes from the received SI and further generate the full video from the reconstructed keyframes using frame interpolation to ensure high-quality reconstruction. Recognizing the impact of wireless channel noise on SI transmission, we also propose an SD-based denoiser for GSC (SD-GSC) condition on an instantaneous channel gain to remove the channel noise from the received noisy SI under a known channel. For scenarios with an unknown channel, we further propose a parallel SD denoiser for GSC (PSD-GSC) to jointly learn the distribution of channel gains and denoise the received SI. It is shown that, with the known channel, our proposed SD-GSC outperforms state-of-the-art ADJSCC, Latent-Diff DNSC, DeepWiVe and DVST, improving Peak Signal-to-Noise Ratio (PSNR) by 69%, 58%, 33% and 38%, reducing mean squared error (MSE) by 52%, 50%, 41% and 45%, and reducing Fréchet Video Distance (FVD) by 38%, 32%, 22% and 24%, respectively. With the unknown channel, our PSD-GSC achieves a 17% improvement in PSNR, a 29% reduction in MSE, and a 19% reduction in FVD compared to MMSE equalizer-enhanced SD-GSC. These significant performance improvements demonstrate the robustness and superiority of our proposed methods in enhancing video transmission quality and efficiency under various channel conditions.

The Fréchet Inception Distance (FID) has been used to evaluate hundreds of generative models. We introduce FastFID, which can efficiently train generative models with FID as a loss function. Using FID as an additional loss for Generative Adversarial Networks improves their FID.

As with many machine learning problems, the progress of image generation methods hinges on good evaluation metrics. One of the most popular is the Frechet Inception Distance (FID). FID estimates the distance between a distribution of Inception-v3 features of real images, and those of images generated by the algorithm. We highlight important drawbacks of FID: Inception's poor representation of the rich and varied content generated by modern text-to-image models, incorrect normality assumptions, and poor sample complexity. We call for a reevaluation of FID's use as the primary quality metric for generated images. We empirically demonstrate that FID contradicts human raters, it does not reflect gradual improvement of iterative text-to-image models, it does not capture distortion levels, and that it produces inconsistent results when varying the sample size. We also propose an alternative new metric, CMMD, based on richer CLIP embeddings and the maximum mean discrepancy distance with the Gaussian RBF kernel. It is an unbiased estimator that does not make any assumptions on the probability distribution of the embeddings and is sample efficient. Through extensive experiments and analysis, we demonstrate that FID-based evaluations of text-to-image models may be unreliable, and that CMMD offers a more robust and reliable assessment of image quality.

We propose the Fréchet Audio Distance (FAD), a novel, reference-free evaluation metric for music enhancement algorithms. We demonstrate how typical evaluation metrics for speech enhancement and blind source separation can fail to accurately measure the perceived effect of a wide variety of distortions. As an alternative, we propose adapting the Fréchet Inception Distance (FID) metric used to evaluate generative image models to the audio domain. FAD is validated using a wide variety of artificial distortions and is compared to the signal based metrics signal to distortion ratio (SDR), cosine distance and magnitude L2 distance. We show that, with a correlation coefficient of 0.52, FAD correlates more closely with human perception than either SDR, cosine distance or magnitude L2 distance, with correlation coefficients of 0.39, -0.15 and -0.01 respectively.

Modern metrics for generative learning like Fréchet Inception Distance (FID) and DINOv2-Fréchet Distance (FD-DINOv2) demonstrate impressive performance. However, they suffer from various shortcomings, like a bias towards specific generators and datasets. To address this problem, we propose the Fréchet Wavelet Distance (FWD) as a domain-agnostic metric based on the Wavelet Packet Transform (W_p). FWD provides a sight across a broad spectrum of frequencies in images with a high resolution, preserving both spatial and textural aspects. Specifically, we use W_p to project generated and real images to the packet coefficient space. We then compute the Fréchet distance with the resultant coefficients to evaluate the quality of a generator. This metric is general-purpose and dataset-domain agnostic, as it does not rely on any pre-trained network, while being more interpretable due to its ability to compute Fréchet distance per packet, enhancing transparency. We conclude with an extensive evaluation of a wide variety of generators across various datasets that the proposed FWD can generalize and improve robustness to domain shifts and various corruptions compared to other metrics.

Fréchet distance has recently emerged as an effective distribution-level objective for generator post-training, complementing the conventional sample-level diffusion and flow-matching losses. However, directly optimizing Fréchet objectives can cause Fréchet hacking. The target metrics keep improving, but visual quality and Fréchet alignment in other feature spaces may stagnate or deteriorate. We attribute this failure to the static pretrained feature spaces used by existing Fréchet losses. These feature spaces provide incomplete and fixed views of the differences between real and generated distributions. To address this limitation, we propose Adversarial Fréchet Distance (AdvFD), which complements the static representation targets in FD-Loss with a calibrated adversarially learned representation. AdvFD augments the original static Fréchet objective with a learnable representation that adversarially maximizes the Fréchet discrepancy between real and generated samples, while the generator minimizes the same discrepancy in the resulting adaptive feature space. To prevent the adversarial representation from trivially increasing the objective through feature amplification, we further introduce real-feature whitening, which normalizes its scale and covariance geometry and stabilizes the min--max optimization. Extensive experiments show that AdvFD consistently improves one-step generator post-training across both JiT and pMF backbones and across different model scales.

Significant advancements have been made in video generative models recently. Unlike image generation, video generation presents greater challenges, requiring not only generating high-quality frames but also ensuring temporal consistency across these frames. Despite the impressive progress, research on metrics for evaluating the quality of generated videos, especially concerning temporal and motion consistency, remains underexplored. To bridge this research gap, we propose Fr\'echet Video Motion Distance (FVMD) metric, which focuses on evaluating motion consistency in video generation. Specifically, we design explicit motion features based on key point tracking, and then measure the similarity between these features via the Fr\'echet distance. We conduct sensitivity analysis by injecting noise into real videos to verify the effectiveness of FVMD. Further, we carry out a large-scale human study, demonstrating that our metric effectively detects temporal noise and aligns better with human perceptions of generated video quality than existing metrics. Additionally, our motion features can consistently improve the performance of Video Quality Assessment (VQA) models, indicating that our approach is also applicable to unary video quality evaluation. Code is available at https://github.com/ljh0v0/FMD-frechet-motion-distance.

Recent progress in Sign Language Translation (SLT) has focussed primarily on improving the representational capacity of large language models to incorporate Sign Language features. This work explores an alternative direction: enhancing the geometric properties of skeletal representations themselves. We propose Geo-Sign, a method that leverages the properties of hyperbolic geometry to model the hierarchical structure inherent in sign language kinematics. By projecting skeletal features derived from Spatio-Temporal Graph Convolutional Networks (ST-GCNs) into the Poincar\'e ball model, we aim to create more discriminative embeddings, particularly for fine-grained motions like finger articulations. We introduce a hyperbolic projection layer, a weighted Fr\'echet mean aggregation scheme, and a geometric contrastive loss operating directly in hyperbolic space. These components are integrated into an end-to-end translation framework as a regularisation function, to enhance the representations within the language model. This work demonstrates the potential of hyperbolic geometry to improve skeletal representations for Sign Language Translation, improving on SOTA RGB methods while preserving privacy and improving computational efficiency. Code available here: https://github.com/ed-fish/geo-sign.

The Frechet Inception Distance (FID) is the de facto arbiter of image generation, yet most papers report just a single number from a single trained model using a single sampling seed. How reproducible is that number if we retrain the model, or merely resample from it? In this paper, we treat FID as a random variable on a two-axis panel of training and generation seeds, and measure its variance directly on several hundred SiT networks trained on class-conditional ImageNet 256x256. We report surprising findings: (a) Retraining the model using the same recipe with a different seed moves FID 3.2x more (in Inception feature space) than redrawing samples from a fixed network. (b) That gap is driven by three factors: random initialisation, data ordering, and the per-step Gaussian noise of the flow-matching loss. (c) Increasing compute or model size barely tightens the spread, holding the FID coefficient of variation (CoV) inside a 1-2% band. (d) Per-cell classifier-free-guidance tuning halves the spread but reshuffles which seeds work best, and a lucky training seed reaches the same FID with up to 2x less compute than an unlucky one. Based on these findings, we recommend a new FID evaluation protocol: evaluate under per-cell optimal guidance, treat any FID gap below the empirically measured ~1.3% CoV as inconclusive, and report an error bar over several training seeds rather than a single FID number.

This paper introduces an end-to-end residual network that operates entirely on the Poincar\'e ball model of hyperbolic space. Hyperbolic learning has recently shown great potential for visual understanding, but is currently only performed in the penultimate layer(s) of deep networks. All visual representations are still learned through standard Euclidean networks. In this paper we investigate how to learn hyperbolic representations of visual data directly from the pixel-level. We propose Poincar\'e ResNet, a hyperbolic counterpart of the celebrated residual network, starting from Poincar\'e 2D convolutions up to Poincar\'e residual connections. We identify three roadblocks for training convolutional networks entirely in hyperbolic space and propose a solution for each: (i) Current hyperbolic network initializations collapse to the origin, limiting their applicability in deeper networks. We provide an identity-based initialization that preserves norms over many layers. (ii) Residual networks rely heavily on batch normalization, which comes with expensive Fr\'echet mean calculations in hyperbolic space. We introduce Poincar\'e midpoint batch normalization as a faster and equally effective alternative. (iii) Due to the many intermediate operations in Poincar\'e layers, we lastly find that the computation graphs of deep learning libraries blow up, limiting our ability to train on deep hyperbolic networks. We provide manual backward derivations of core hyperbolic operations to maintain manageable computation graphs.

We show that Fréchet Distance (FD), long considered impractical as a training objective, can in fact be effectively optimized in the representation space. Our idea is simple: decouple the population size for FD estimation (e.g., 50k) from the batch size for gradient computation (e.g., 1024). We term this approach FD-loss. Optimizing FD-loss reveals several surprising findings. First, post-training a base generator with FD-loss in different representation spaces consistently improves visual quality. Under the Inception feature space, a one-step generator achieves0.72 FID on ImageNet 256x256. Second, the same FD-loss repurposes multi-step generators into strong one-step generators without teacher distillation, adversarial training or per-sample targets. Third, FID can misrank visual quality: modern representations can yield better samples despite worse Inception FID. This motivates FDr^k, a multi-representation metric. We hope this work will encourage further exploration of distributional distances in diverse representation spaces as both training objectives and evaluation metrics for generative models.

Contrast agents in dynamic contrast enhanced magnetic resonance imaging allow to localize tumors and observe their contrast kinetics, which is essential for cancer characterization and respective treatment decision-making. However, contrast agent administration is not only associated with adverse health risks, but also restricted for patients during pregnancy, and for those with kidney malfunction, or other adverse reactions. With contrast uptake as key biomarker for lesion malignancy, cancer recurrence risk, and treatment response, it becomes pivotal to reduce the dependency on intravenous contrast agent administration. To this end, we propose a multi-conditional latent diffusion model capable of acquisition time-conditioned image synthesis of DCE-MRI temporal sequences. To evaluate medical image synthesis, we additionally propose and validate the Fréchet radiomics distance as an image quality measure based on biomarker variability between synthetic and real imaging data. Our results demonstrate our method's ability to generate realistic multi-sequence fat-saturated breast DCE-MRI and uncover the emerging potential of deep learning based contrast kinetics simulation. We publicly share our accessible codebase at https://github.com/RichardObi/ccnet and provide a user-friendly library for Fréchet radiomics distance calculation at https://pypi.org/project/frd-score.