论文检索

Theory · Reinforcement Learning and Planning

Patrik Reizinger, Bálint Mucsányi, Siyuan Guo, Benjamin Eysenbach, Bernhard Schölkopf, Wieland Brendel

Self-supervised feature learning and pretraining methods in reinforcement learning (RL) often rely on information-theoretic principles, termed mutual information skill learning (MISL). These methods aim to learn a representation of the environment while also incentivizing exploration thereof. However, the role of the representation and mutual information parametrization in MISL is not yet well understood theoretically. Our work investigates MISL through the lens of identifiable representation learning by focusing on the Contrastive Successor Features (CSF) method. We prove that CSF can provably recover the environment's ground-truth features up to a linear transformation due to the inner product parametrization of the features and skill diversity in a discriminative sense. This first identifiability guarantee for representation learning in RL also helps explain the implications of different mutual information objectives and the downsides of entropy regularizers. We empirically validate our claims in MuJoCo and DeepMind Control, and show that CSF provably recovers the ground-truth features from both states and pixels. Our code is available at https://github.com/bmucsanyi/identifiable-misl.

Social Aspects · Accountability, Transparency and Interpretability

Yuhan Guo, Lizhong Ding, Shihao Jia, Yanyu Ren, Pengqi Li, Jiarun Fu, Changsheng Li, Ye Yuan, Guoren Wang

\textit{Learning to Explain} is a forward-looking paradigm recently proposed in the field of explainable AI, which envisions training explainers capable of producing high-quality explanations for target models efficiently. Although existing studies have made attempts through self-supervised optimization or learning from prior explanation methods, the \textit{Learning to Explain} paradigm still faces three critical challenges: 1) self-supervised objectives rely on assumptions about the target model or task, restricting their generalizability; 2) methods driven by prior explanations struggle to guarantee the quality of the supervisory signals; and 3) depending exclusively on either approach leads to poor convergence or limited explanation quality. To address these challenges, we propose a \textit{faithfulness}-guided amortized explainer that 1) theoretically derives a self-supervised objective free from assumptions about the target model or task, 2) practically generates high-quality supervisory signals by deduplicating and filtering prior explanations, and 3) jointly optimizes both objectives via a dynamic weighting strategy, enabling the amortized explainer to produce more faithful explanations for complex, high-dimensional models. We re-formalize multiple well-validated faithfulness evaluation metrics within a unified notation system and theoretically prove that an explanation mapping can simultaneously achieve optimality across all these metrics. We aggregate prior explanation methods to generate high-quality supervised signals through deduplicating and faithfulness-based filtering. Our amortized explainer leverages dynamic weighting to guide optimization, initially emphasizing pattern consistency with the supervised signals for rapid convergence, and subsequently refining explanation quality by approximating the most faithful explanation mapping. Extensive experiments across various target models and image, text, and tabular tasks demonstrate that the proposed explainer consistently outperforms all prior explanation methods across all faithfulness metrics, highlighting its effectiveness and its potential to offer a systematic solution to the fundamental challenges of the \textit{Learning to Explain} paradigm.

Computer Vision · Vision Models & Multimodal

Haonan Wang, Hongfu Liu, Xiangyan Liu, Chao Du, Kenji Kawaguchi, Ye Wang, Tianyu Pang

Next-token prediction serves as the foundational learning task that enables reasoning in LLMs. But what should the learning task be when aiming to equip MLLMs with temporal reasoning capabilities over video inputs? Existing tasks such as video captioning primarily promote modality alignment, while video question answering typically relies on annotations from humans or much stronger MLLMs. To address this gap, we propose next-event prediction (NEP), a learning task that harnesses future video segments as a rich, self-supervised signal to foster temporal reasoning. We segment each video into past and future frames: the MLLM takes the past frames as input and predicts events in the future, thereby encouraging the model to reason temporally in order to complete the task. To study this learning task, we curate V1-33K, a dataset comprising 33,000 automatically extracted videos spanning diverse real-world scenarios. Using the same videos, we further explore a range of video instruction-tuning tasks data to provide controlled comparisons and isolate the effect of NEP. To evaluate progress, we introduce FutureBench to assess coherence in predicting unseen future events. Experiments validate that NEP offers a scalable and effective training task for fostering temporal reasoning in MLLMs.

Applications · Neuroscience, Cognitive Science

Yanchen Wang, Han Yu, Ari Blau, Yizi Zhang, International Brain Laboratory, Liam Paninski, Cole Hurwitz, Matthew R Whiteway

The brain can only be fully understood through the lens of the behavior it generates--a guiding principle in modern neuroscience research that nevertheless presents significant technical challenges. Many studies capture behavior with cameras, but video analysis approaches typically rely on specialized models requiring extensive labeled data. We address this limitation with BEAST (BEhavioral Analysis via Self-supervised pretraining of Transformers), a novel and scalable framework that pretrains experiment-specific vision transformers for diverse neuro-behavior analyses. BEAST combines masked autoencoding with temporal contrastive learning to effectively leverage unlabeled video data. Through comprehensive evaluation across multiple species, we demonstrate improved performance in three critical neuro-behavioral tasks: extracting behavioral features that correlate with neural activity, and pose estimation and action segmentation in both the single- and multi-animal settings. Our method establishes a powerful and versatile backbone model that accelerates behavioral analysis in scenarios where labeled data remains scarce.

Yuhang Chen, Zhen Tan, Ruichen Zhang, Mufan Qiu, Tianlong Chen

\begin{abstract} Developing generalist large language models (LLMs) capable of complex biological reasoning is a central challenge in computational biology. While existing LLMs excel at predictive tasks like cell type annotation and logically-constrained problems, enabling open-ended and mechanistic reasoning remains a challenge. A promising direction is Reinforcement Learning from Verifiable Rewards (RLVR), which has been shown to significantly enhance complex reasoning in general domains like mathematics and code synthesis. However, its application in biology is hindered, as most biological outcomes are non-verifiable. For example, verifying a generated gene sequence is usually infeasible. In this paper, we introduce CellDuality, a self-supervised framework that enables LLM agents for robust reasoning in single-cell biology. Our framework is built on the principle of complementary task duality, a self-verification process that leverages a bidirectional reasoning loop. First, the model performs a forward reasoning task by predicting a biological outcome (e.g., a cell's response to a drug). Then, in a complementary inverse task, it must reason backward from its own prediction to reconstruct the initial conditions (e.g., the original drug perturbation). The fidelity of this reconstruction serves as an intrinsic reward signal, creating a feedback loop that enforces logical and biological consistency. We use these intrinsic rewards to align the base LLM via reinforcement learning, without requiring ground-truth verification labels. We demonstrate that CellDuality achieves state-of-the-art performance and provides coherent biological explanations across a diverse suite of single-cell reasoning tasks. Critically, on the challenging out-of-distribution perturbation prediction benchmark, our self-supervised approach significantly outperforms the standard fine-tuning baseline and narrows the performance gap to a supervised RLVR baseline. Our work showcases a new path toward scalable training of biological foundation models.

Computer Vision · Vision Models & Multimodal

Anthony Fuller, Yousef Yassin, Daniel Kyrollos, Evan Shelhamer, James Green

ViTs are general and accurate, and address many tasks, but ViTs are slow, and are not always practical when efficiency is key. Existing methods for faster ViTs design hybrid non-ViT architectures, losing generality, or shrink their tokens, sacrificing accuracy. Many non-ViT architectures are both fast and accurate. Yet, without significant modifications, they cannot do what ViTs can: process other input shapes, pre-train by SOTA self-supervised learning, reduce computation by dropping tokens, and more. We make ViTs faster by reducing patch token width while increasing global token width by adding a new Jumbo token. Our wider Jumbo token is processed by its own wider FFN to increase model capacity. Yet our Jumbo FFN is efficient: it processes a single token, for speed, and its parameters are shared across all layers, for memory. Crucially, our Jumbo is attention-only and non-hierarchical, like a plain ViT, so it is simple, scalable, flexible, and compatible with ViT methods new and old. Jumbo improves over ViT baselines with Registers from Nano to Large scales while maintaining speed/throughput on ImageNet-1K (0.1-13%). Jumbo also improves segmentation (1.9-3.1% on ADE20K), MAE pre-training (4.9% linear probing on ImageNet-1K), test-time adaptation (5.2% on ImageNet-C), and time series modeling. Our Jumbo models even achieve better speed-accuracy trade-offs than specialized non-ViT compute-efficient models, while maintaining plain-ViT compatibility for practicality. Code and weights are available: https://github.com/antofuller/jumbo

Computer Vision · Image and Video Generation

Matthieu Terris, Samuel Hurault, Maxime Song, Julián Tachella

Most existing learning-based methods for solving imaging inverse problems can be roughly divided into two classes: iterative algorithms, such as plug-and-play and diffusion methods leveraging pretrained denoisers, and unrolled architectures that are trained end-to-end for specific imaging problems. Iterative methods in the first class are computationally costly and often yield suboptimal reconstruction performance, whereas unrolled architectures are generally problem-specific and require expensive training. In this work, we propose a novel non-iterative, lightweight architecture that incorporates knowledge about the forward operator (acquisition physics and noise parameters) without relying on unrolling. Our model is trained to solve a wide range of inverse problems, such as deblurring, magnetic resonance imaging, computed tomography, inpainting, and super-resolution, and handles arbitrary image sizes and channels, such as grayscale, complex, and color data. The proposed model can be easily adapted to unseen inverse problems or datasets with a few fine-tuning steps (up to a few images) in a self-supervised way, without ground-truth references. Throughout a series of experiments, we demonstrate state-of-the-art performance from medical imaging to low-photon imaging and microscopy. Our code is available at https://github.com/matthieutrs/ram.

Computer Vision · Image and Video Generation

Minglei Shi, Haolin Wang, Wenzhao Zheng, Ziyang Yuan, Xiaoshi Wu, Xintao WANG, Pengfei Wan, Jie Zhou, Jiwen Lu

Recent progress in diffusion-based visual generation has largely relied on latent diffusion models with Variational Autoencoders (VAEs). While effective for high-fidelity synthesis, this VAE+Diffusion paradigm still suffers from limited training and inference efficiency, along with poor transferability to broader vision tasks. These issues stem from a key limitation of VAE latent spaces: the lack of clear semantic separation and strong discriminative structure. Our analysis confirms that these properties are not only crucial for perception and understanding tasks, but also equally essential for the stable and efficient training of latent diffusion models. Motivated by this insight, we introduce **SVG**—a novel latent diffusion model without variational autoencoders, which unleashes **S**elf-supervised representations for **V**isual **G**eneration. SVG constructs a feature space with clear semantic discriminability by leveraging frozen DINO features, while a lightweight residual branch captures fine-grained details for high-fidelity reconstruction. Diffusion models are trained directly on this semantically structured latent space to facilitate more efficient learning. As a result, SVG enables accelerated diffusion training, supports few-step sampling, and improves generative quality. Experimental results further show that SVG preserves the semantic and discriminative capabilities of the underlying self-supervised representations, providing a principled pathway toward task-general, high-quality visual representations.

Deep Learning · Everything Else

Yiping Ji, James Martens, Jianqiao Zheng, Ziqin Zhou, Peyman Moghadam, Xinyu Zhang, Hemanth Saratchandran, Simon Lucey

Transformers have achieved remarkable success across a wide range of applications, a feat often attributed to their scalability. Yet training them without residual (skip) connections remains notoriously difficult. While skips stabilize optimization, they also disrupt the hierarchical structure of representations, raising the long-standing question of whether transformers can be trained efficiently without them. In this work, we address this problem by analyzing the Jacobian of a skipless transformer block, showing why residuals improve conditioning and revealing that their stabilization benefits can be recovered through a principled initialization strategy. Building on this insight, we introduce the first method that enables stable and efficient training of skipless transformers without altering the standard architecture. We validate our approach on Vision Transformers (ViTs) in both supervised and self-supervised settings, demonstrating that skipless ViTs trained with our initialization overcome the usual optimization barriers, learn richer hierarchical representations, and outperform strong residual baselines on dense prediction benchmarks. These results show that skip connections are not a fundamental requirement for training ViTs and open new avenues for hierarchical representation learning in vision models.

Applications · Robotics

Yingyan Li, Shuyao Shang, Weisong Liu, Bing Zhan, Haochen Wang, Yuqi Wang, Yuntao Chen, Xiaoman Wang, Yasong An, Chufeng Tang 等

Scaling Vision-Language-Action (VLA) models on large-scale data offers a promising path to achieving a more generalized driving intelligence. However, VLA models are limited by a ``supervision deficit'': the vast model capacity is supervised by sparse, low-dimensional actions, leaving much of their representational power underutilized. To remedy this, we propose DriveVLA-W0, a training paradigm that employs world modeling to predict future images. This task generates a dense, self-supervised signal that compels the model to learn the underlying dynamics of the driving environment. We showcase the paradigm's versatility by instantiating it for two dominant VLA archetypes: an autoregressive world model for VLAs that use discrete visual tokens, and a diffusion world model for those operating on continuous visual features. Building on the rich representations learned from world modeling, we introduce a lightweight action expert to address the inference latency for real-time deployment. Extensive experiments on the NAVSIM v1/v2 benchmark and a 680x larger in-house dataset demonstrate that DriveVLA-W0 significantly outperforms BEV and VLA baselines. Crucially, it amplifies the data scaling law, showing that performance gains accelerate as the training dataset size increases.

General Machine Learning · Representation Learning

Jean Ponce, Basile Terver, Martial Hebert, Michael Arbel

The {\em stop gradient} and {\em exponential moving average} iterative procedures are commonly used in non-contrastive approaches to self-supervised learning to avoid representation collapse, with excellent performance in downstream applications in practice. This presentation investigates these procedures from the dual viewpoints of optimization and dynamical systems. We show that, in general, although they {\em do not} optimize the original objective, or {\em any} other smooth function, they {\em do} avoid collapse. Following [Tian et al. 2021], but without any of the extra assumptions used in their proofs, we then show using a dynamical system perspective that, in the linear case, minimizing the original objective function without the use of a stop gradient or exponential moving average {\em always} leads to collapse. Conversely, we characterize explicitly the equilibria of the dynamical systems associated with these two procedures in this linear setting as algebraic varieties in their parameter space, and show that they are, in general, {\em asymptotically stable}. Our theoretical findings are illustrated by empirical experiments with real and synthetic data.

Ru Wang, Wei Huang, Qi Cao, Yusuke Iwasawa, Yutaka Matsuo, Jiaxian Guo

Test-time reinforcement learning (TTRL) offers a label-free paradigm for adapting models using only synthetic signals at inference, but its success hinges on constructing reliable learning signals. Standard approaches such as majority voting often collapse to spurious yet popular answers. We introduce Self-Harmony, a framework built on a simple intuition: the correct answer should remain stable across both an original question and its paraphrase. Self-Harmony operationalizes this by employing a single model in two complementary roles: a Solver to produce answers and a Reframer to rephrase the input. Based on this, we further propose a pseudo-label method: instead of majority voting, it aggregates answer frequencies across these original and reframed views using the harmonic mean. This is a process that naturally selects for solutions stable under reframing, thereby avoiding the common trap of favoring view-dependent, spurious answers. Crucially, this requires no human supervision or auxiliary models. Across diverse reasoning benchmarks, Self-Harmony achieves state-of-the-art results at the label-free test-time setting, ranking first in 28 of 30 settings across multiple methods. Beyond accuracy, it demonstrates unprecedented robustness, with zero training failures in all experiments, underscoring its stability and reliability.

Lvda Chen, Yang Li, Junchi Yan

Neural Combinatorial Optimization (NCO) has long been anchored in paradigms such as solution construction or improvement that treat the solution as a monolithic reference, squandering the rich local decision patterns embedded in high-quality solutions. Inspired by the scalability of self-supervised pretraining in language and vision, we propose a shift in perspective: Can combinatorial optimization adopt a fundamental training paradigm to enable scalable representation learning? We introduce MaskCO, a masked generation approach that reframes learning to optimize as self-supervised learning on given reference solutions. By strategically masking portions of optimal solutions and training models to recover the missing content, MaskCO turns a single instance-solution pair into a multitude of local learning signals, forcing the model to internalize fine-grained structural dependencies. At inference time, we employ a mask-and-reconstruct procedure, i.e., a refinement loop that iteratively masks variables and regenerates them to progressively improve solution quality. Our findings show that these learned representations are highly transferable, facilitating effective fine-tuning and boosting the performance of alternative inference approaches. Experimental results demonstrate that MaskCO achieves remarkable performance improvements over previous state-of-the-art neural solvers, reducing the optimality gap by more than 99% and achieving a 10x speedup on problems such as the Travelling Salesman Problem (TSP).

Applications · Neuroscience, Cognitive Science

Yiting Dong, Jianhao Ding, Zijie Xu, Tong Bu, Zhaofei Yu, Tiejun Huang

Spiking Neural Networks (SNNs), with their temporal processing capabilities and biologically plausible dynamics, offer a natural platform for unsupervised representation learning. However, current unsupervised SNNs predominantly employ shallow architectures or localized plasticity rules, limiting their ability to model long-range temporal dependencies and maintain temporal feature consistency. This results in semantically unstable representations, thereby impeding the development of deep unsupervised SNNs for large-scale temporal video data. We propose PredNext, which explicitly models temporal relationships through cross-view future Step Prediction and Clip Prediction. This plug-and-play module seamlessly integrates with diverse self-supervised objectives. We firstly establish standard benchmarks for SNN self-supervised learning on UCF101, HMDB51, and MiniKinetics, which are substantially larger than conventional DVS datasets. PredNext delivers significant performance improvements across different tasks and self-supervised methods. PredNext achieves performance comparable to ImageNet-pretrained supervised weights, through unsupervised training solely on UCF101. Additional experiments demonstrate that PredNext, distinct from forced consistency constraints, substantially improves temporal feature consistency while enhancing network generalization capabilities. This work provides a effective foundation for unsupervised deep SNNs on large-scale temporal video data.

Theory · Interpretability and Visualization

Usha Bhalla, Alex Oesterling, Claudio Mayrink Verdun, Hima Lakkaraju, Flavio Calmon

Translating the internal representations and computations of models into concepts that humans can understand is a key goal of interpretability. While recent dictionary learning methods such as Sparse Autoencoders (SAEs) provide a promising route to discover human-interpretable features, they often only recover token-specific, noisy, or highly local concepts. We argue that this limitation stems from neglecting the temporal structure of language, where semantic content typically evolves smoothly over sequences. Building on this insight, we introduce Temporal Sparse Autoencoders (T-SAEs), which incorporate a novel contrastive loss encouraging consistent activations of high-level features over adjacent tokens. This simple yet powerful modification enables SAEs to disentangle semantic from syntactic features in a self-supervised manner. Across multiple datasets and models, T-SAEs recover smoother, more coherent semantic concepts without sacrificing reconstruction quality. Strikingly, they exhibit clear semantic structure despite being trained without explicit semantic signal, offering a new pathway for unsupervised interpretability in language models.

Computer Vision · Vision Models & Multimodal

Wen Wen, Tianwu Zhi, Kanglong Fan, Yang Li, Xinge Peng, Yabin ZHANG, Yiting Liao, Junlin Li, Li zhang

Improving vision-language models (VLMs) in the post-training stage typically relies on supervised fine-tuning or reinforcement learning, methods that necessitate costly, human-annotated data. While self-supervised techniques such as self-consistency have proven effective for enhancing reasoning capabilities, their application to perceptual domains such as image quality assessment (IQA) remains largely unexplored. In this work, we introduce EvoQuality, a novel framework that enables a VLM to autonomously refine its quality perception capabilities without any ground-truth labels. EvoQuality adapts the principle of self-consistency to the ranking-based nature of IQA. It generates pseudo-labels by performing pairwise majority voting on the VLM's own outputs to establish a consensus on relative quality. These pseudo-rankings are then formulated into a fidelity reward that guides the model's iterative evolution through group relative policy optimization (GRPO). By iteratively leveraging its own predictions, EvoQuality progressively refines the VLM’s perceptual capability. Extensive experiments show that EvoQuality boosts the base VLM's zero-shot performance by 31.8% on PLCC across diverse IQA benchmarks. Remarkably, despite being entirely self-supervised, EvoQuality achieves performance that is competitive with, or even surpasses, state-of-the-art supervised VLM-based IQA models, outperforming these models on 5 out of 7 IQA benchmarks. Furthermore, the framework demonstrates significant flexibility, allowing it to be stacked with pre-trained IQA models to bolster generalization on unseen datasets.

General Machine Learning · Transfer, Multitask and Meta-learning

Kunal Pratap Singh, Ali Garjani, Rishubh Singh, Muhammad Uzair Khattak, Jason Toskov, Efe Tarhan, Andrei Atanov, Oğuzhan Kar, Amir Zamir

The common approach for developing a vision model is generalism, which involves training on a large diverse dataset to cover the varied deployment environments and leads to a model that is expected to solve the problem everywhere. However, many practical applications need to operate in a specific test space, e.g., a robot deployed in a single house, and do not necessarily need to generalize to novel environments. In this work, we explore whether we can use rich multimodal data only from the test environment to pre-train a representation in a self-supervised way, without access to any external data. We find that this approach can match and, in most cases, outperform generalists pre-trained on large-scale Internet datasets, including popular off-the-shelf models, CLIP and DINOv2. We study the effectiveness of this approach by evaluating the models on various datasets and downstream tasks, such as semantic segmentation, captioning, and object detection, as well as a set of ablations and analyses to extract insights. This approach raises intriguing points on substituting data with (multi)modality, enabling an alternative scenario where the need for external Internet-scale datasets for pre-training models is reduced. It also shows that merely benefiting from test-space data was insufficient for achieving competitive results, and multimodality was essential for that purpose.

Applications · Chemistry and Drug Discovery

Giacomo Turri, Luigi Bonati, Kai Zhu, massimiliano pontil, Pietro Novelli

We introduce an end-to-end approach to learn the evolution operators of large-scale non-linear dynamical systems, such as those describing complex natural phenomena. Evolution operators are particularly well-suited for analyzing systems that exhibit spatio-temporal patterns and have become a key analytical tool across various scientific communities. As terabyte-scale weather datasets and simulation tools capable of running millions of molecular dynamics steps per day are becoming commodities, our approach provides an effective tool to make sense of them from a data-driven perspective. The core of it lies in a remarkable connection between self-supervised representation learning methods and the recently established learning theory of evolution operators. We deploy our approach across multiple scientific domains: explaining the folding dynamics of small proteins, the binding process of drug-like molecules in host sites, and autonomously finding patterns in climate data. Our code is available open-source at: https://github.com/pietronvll/encoderops.

Applications · Neuroscience, Cognitive Science

Ling Zhan, Zhen Li, Junjie Huang, Tao Jia

Benchmarking the hundreds of functional connectivity (FC) modeling methods on large-scale fMRI datasets is critical for reproducible neuroscience. However, the combinatorial explosion of model–data pairings makes exhaustive evaluation computationally prohibitive, preventing such assessments from becoming a routine pre-analysis step. To break this bottleneck, we reframe the challenge of FC benchmarking by selecting a small, representative *core-set* whose sole purpose is to preserve the relative performance ranking of FC operators. We formalize this as a ranking-preserving subset selection problem and propose **S**tructure-aware **C**ontrastive **L**earning for **C**ore-set **S**election (**SCLCS**), a self-supervised framework to select these core-sets. **SCLCS** first uses an adaptive Transformer to learn each sample's unique FC structure. It then introduces a novel **S**tructural **P**erturbation **S**core (**SPS**) to quantify the stability of these learned structures during training, identifying samples that represent foundational connectivity archetypes. Finally, while **SCLCS** identifies stable samples via a top-$k$ ranking, we further introduce a **density-balanced sampling strategy** as a necessary correction to promote diversity, ensuring the final core-set is both structurally robust and distributionally representative. On the large-scale REST-meta-MDD dataset, **SCLCS** preserves the ground-truth model ranking with just 10% of the data, outperforming state-of-the-art (SOTA) core-set selection methods by up to 23.2% in ranking consistency (nDCG@k). To our knowledge, this is the first work to formalize core-set selection for FC operator benchmarking, thereby making large-scale operators comparisons a feasible and integral part of computational neuroscience. Code is publicly available on: [https://github.com/lzhan94swu/SCLCS](https://github.com/lzhan94swu/SCLCS)

Deep Learning · Generative Models and Autoencoders

Victor Sechaud, Jérémy Scanvic, Quentin Barthélemy, Patrice Abry, Julián Tachella

Self-supervised learning for inverse problems allows to train a reconstruction network from noise and/or incomplete data alone. These methods have the potential of enabling learning-based solutions when obtaining ground-truth references for training is expensive or even impossible. In this paper, we propose a new self-supervised learning strategy devised for the challenging setting where measurements are observed via a single incomplete observation model. We introduce a new definition of equivariance in the context of reconstruction networks, and show that the combination of self-supervised splitting losses and equivariant reconstruction networks results in unbiased estimates of the supervised loss. Through a series of experiments on image inpainting, accelerated magnetic resonance imaging, sparse-view computed tomography, and compressive sensing, we demonstrate that the proposed loss achieves state-of-the-art performance in settings with highly rank-deficient forward models.