Reconstruction Error

We study reconstruction in autoencoders that apply the same forward map before and after setting the observed coordinates to zero. For equal odd input and hidden dimensions $d\geq 3$, among orientation-preserving diffeomorphisms whose Jacobian singular values lie in $[m,M]$, we show that the least uniform reconstruction-derivative error is $\max\{1-M(M-m)/2,0\}$, with affine maps attaining this sharp bound at every prescribed depth. A translated radial rotation can nevertheless reconstruct any prescribed ball exactly with singular values arbitrarily close to one, motivating additional conditions for a finite-data bound. We test this prediction on a 798,452-point terrestrial LiDAR forest scan. At input scale $0.05$, the mean theoretical bound is $0.155$, about $84\%$ of the mean normalized training error $0.185$ across four spatial regions, two depths, and three seeds. At this scale, adding one hidden coordinate reduces the mean reconstruction error below $6\times10^{-6}$.

Weight-only post-training quantization is the cheapest way to shrink a retrieval embedder, and the received advice for applying it -- protect the embedding table, allocate bits by module sensitivity, prefer a ranking-aware objective over weight reconstruction -- was carried into LLM quantization largely intact. We test that advice on retrieval embedders directly, quantizing five checkpoints from four architecture families across a grid of bit widths and group sizes, and isolating the embedding, attention and feed-forward blocks at each width. Every heuristic fails to transfer as stated. The embedding table never emerges as the dominant isolated protection priority in any family, despite being the largest tensor in several of them. Module sensitivity does not survive as a transferable ordering: at INT4/g16 the spread between modules is too small to allocate against, at INT3 the ordering becomes family-dependent and joint damage stops being the sum of its parts, and at INT2 comparable reconstruction error accompanies retention ranging from 1.3 to 65.9 percent of full precision. A cheap reconstruction proxy is useful for screening uniform bit widths but substantially less reliable for choosing which tensors to protect; its apparent strength across the whole grid is a range-extension artifact. A distilled 109M student at INT3 holds 78.04 NDCG@10 in 68.4 MB and dominates the extreme-PTQ arm of its own 0.6B teacher, 297.9 MB at 64.46, on both size and quality -- but only inside the task it was distilled for. Sizes are byte counts of files that exist rather than arithmetic estimates, and the measurement repository carries the byte provenance for every one of them.

Self-supervised pre-training via cross-view completion learns strong features for 3D vision from co-visible regions of image pairs. However, the reference view provides little information for reconstructing non-co-visible patches, implicitly yielding a monocular training signal in these regions. We introduce Gekko, which turns this limitation into a useful signal. The relative improvement of the cross-view reconstruction error over a masked-autoencoder error is a self-supervised proxy for co-visibility: large improvements indicate co-visible regions, negligible ones non-co-visible areas. Gekko is a network, trained from scratch, that jointly performs cross-view completion, masked autoencoding, and per-pixel prediction of this relative improvement, providing an additional binocular signal for all masked regions without any ground-truth 3D annotation. Under identical architectures and training data, Gekko consistently outperforms CroCo on zero-shot correspondence estimation, relative pose estimation, and pointmap regression, with up to 6 times higher accuracy at the strictest relative-pose threshold and a 22% drop in end-point error on ETH3D. The extra channel it learns is itself a strong co-visibility detector on unseen scenes, and Gekko's frozen features outperform released cross-view backbones of comparable or larger size. It can also be trained directly from raw videos with a simple stride-based curriculum, removing the cumbersome 3D preprocessing prior methods require while matching models trained on curated data. Code and pre-trained models are publicly available.

Neural PDE training yields a finite checkpoint archive, yet its logged energy errors are inaccessible without the exact solution, while loss-based selection does not necessarily recover the logged energy oracle. For admissible neural approximations of symmetric coercive variational problems, we introduce a reference-free selection rule based on minimizing a computable conforming Riesz monitor. The exact residual-energy identity and conforming projection make the monitor an unconditional lower bound converging monotonically to each logged energy error under nested conforming refinement; under saturation, hierarchical enrichment yields a computable upper estimate and hence a lower-upper bracket. A key finding is that archive selection is order-sensitive: unresolved checkpoint-dependent components can reverse the oracle-non-oracle ranking at finite resolution, so checkpointwise recovery alone is insufficient. For finite archives, we prove uniform recovery, yielding convergence to the logged-oracle error and, without saturation, logged-oracle selection at sufficiently fine auxiliary resolution. Under saturation, the bracket gives a computable near-oracle bound and certifies unique logged-oracle selection upon interval separation. We also bound logging-resolution loss and certify oracle inclusion over prescribed comparison trajectories. The resulting criterion replaces inaccessible exact-error minimization by computable, training-independent post-training selection on the intrinsic energy-error scale, requiring only the computed candidates and the variational problem. Experiments on diffusion and elasticity, including a non-manufactured perforated plate, demonstrate energy-scale calibration, oracle-level selection, and modest post-processing cost.

Event cameras, with high temporal resolution, high dynamic range, and asynchronous sensing characteristics, have shown great potential for dense 3D reconstruction. Traditional reconstruction methods based on off-the-shelf pose estimates achieve high efficiency but produce low-fidelity results, as inaccurate pose initialization introduces cumulative reconstruction errors. In contrast, recent SLAM-style methods stabilize joint pose-scene optimization through incremental tracking and mapping, yielding higher reconstruction fidelity at the expense of considerable computational overhead. To address this trade-off, this paper presents EvTrajGS, an accurate and efficient 3D Gaussian Splatting framework for unposed event streams. Our method enables reliable joint pose-scene optimization initialized from coarse pose priors, eliminating the need for computationally expensive SLAM-style pipelines. EvTrajGS parameterizes camera motion as a continuous-time trajectory initialized from discrete camera poses, providing a unified representation for pose refinement. We then aggregate adjacent trajectory states into a temporally coupled pose, promoting temporally consistent pose updates during joint optimization. Additionally, we introduce a loss-reweighted event sampling strategy to adaptively emphasize temporally under-reconstructed intervals. Extensive experiments on both synthetic and real-world datasets demonstrate that EvTrajGS outperforms state-of-the-art methods in terms of both geometric reconstruction quality and pose estimation accuracy, achieving 3.8 dB higher PSNR, 0.1 higher SSIM, and over 40% lower ATE RMSE while retaining high computational efficiency.

Latent low-dimensional structure in datasets of natural and engineered systems enables their sparse sensing, or full-state reconstruction from historical data and very few carefully chosen localized measurements. Depending on the reconstruction algorithm, sensor locations, and measurement noise, the reconstruction risk curves demonstrate a diversity of patterns including a dramatic peak in error known as double descent in Machine Learning literature. Here we explore those scenarios under a unified Data-Noise Averaging theory. Qualitatively, we formulate sufficient criteria for double descent to emerge through a catastrophic amplification of a pathological signal in reconstruction. Quantitatively, we predict the detailed risk curves at a fraction of computational cost, trace reconstruction instability to individual sensors and their combinations, and provide regularization mechanisms to mitigate the instability. We demonstrate results for both static reconstruction of Sea Surface Temperature patterns and time integration of a reduced order model of a PDE.

Matricized low-rank approximation via SVD is a standard surrogate for tensor decompositions, but entry-wise reconstruction error fails to capture multiway geometric degradation. Under an orthogonal Tucker model, we characterize this degradation using two metrics: cross-mode Direction Loss, measuring geometric subspace deviation from rank truncation and noise rotation, and Interaction Loss, quantifying multilinear interaction distortion in the core tensor. We prove that squared relative reconstruction error orthogonally decomposes into interaction loss and out-of-subspace energy, and derive a Wedin-type bound establishing the stability of a plug-in Direction Loss estimator. Experiments on synthetic and hyperspectral datasets demonstrate that nearly identical reconstruction errors can yield markedly different structural-loss profiles; hyperspectral patches with comparable reconstruction errors exhibit up to a 4.6-fold difference in Direction Loss, correlating with severe visual blurring.

Smart-building load forecasters are often trained offline on dense, multivariate, high-frequency data, but deployment may provide only hourly, feature-limited inputs. Missing features must then be reconstructed, and their errors can propagate through the model. If this input uncertainty is not reflected, prediction intervals may become miscalibrated, affecting demand-response scheduling. Our work examines where uncertainty should be placed once inference inputs are reconstructed. We develop a unified one-day-ahead probabilistic forecasting framework that aligns temporal resolution, reconstructs the unavailable inputs, and derives causal features, and we compare a modular post-hoc residual-quantile scheme with an integrated in-model quantile-learning scheme. The comparison uses three mid-scale Deep Learning (DL) backbones: recurrent, hybrid recurrent, and attention-based Temporal Fusion Transformer (TFT) models, under identical inputs, forecasting horizon, preprocessing rules, and training budgets. Results show that uncertainty placement is backbone-dependent. Integrated quantile learning is most reliable with the TFT, yielding 2.2-3.6% MAPE and 28-83W RMSE on the labeled test window, while producing intervals about 5x narrower than the modular intervals at the closest-to-nominal coverage level. Diebold-Mariano tests support the TFT ranking and the mixed behavior of the recurrent backbones. A reconstruction-sensitivity test shows that reconstructed inputs increase the Quantile Score (QS) by 106% while interval width remains nearly unchanged, indicating that the model does not automatically absorb reconstruction-induced uncertainty. Robustness checks against non-DL baselines and seasonal hold-out weeks support this ranking. Our results expose the limits of post-hoc residual quantiles when inference depends on reconstructed inputs.

We study kernel-based operator learning in a two-stage sampling framework, where an offline kernel regression operator learns a discretized representation of the target operator from input-output pairs and an online kernel reconstruction operator recovers the output function from predicted observations. Our main theoretical contribution is an explicit budget allocation condition relating the number $N$ of training pairs, the number $n$ of input observations, and the output resolution $m$. The condition is derived from a coupled error analysis that interprets the surrogate as a reconstruction from approximate data. This yields a decomposition of the total error into reconstruction and learning contributions that can be analyzed independently. As a consequence, we obtain quantitative scaling laws describing how $N$, $n$, and $m$ must be coupled to guarantee convergence and to balance offline learning and online reconstruction errors. The resulting estimates extend previous analyses of kernel-based operator learning. We further introduce a physics-informed extension that incorporates knowledge of the underlying PDE at evaluation time. Rather than encoding constraints directly into the kernel, we augment the online reconstruction step by penalizing PDE residuals at collocation points. The method requires no retraining for new inputs. Numerical experiments illustrate the theoretical findings and demonstrate the effectiveness of the proposed physics-informed reconstruction strategy.

In X-ray tomography, reconstruction quality generally improves with larger numbers of projections. However, more projections increase experiment costs, acquisition time and the radiation dose imparted to the sample. One mitigation to these trade-offs is to adopt a sequential design of experiments, in which each subsequent measurement is determined as a function of previously acquired data in order to maximize information gain. In practice, a widely used heuristic to maximize information is to align beams with the edges of the sample. A key challenge, however, is that the true sample is unknown, so identifying edge-aligned beams typically requires reconstructing the sample based on available measurements. This work proposes a novel sequential design method that identifies edge-aligned measurements directly from the sinogram, bypassing any reconstruction, thereby improving computational efficiency and reducing the experimental design's susceptibility to reconstruction errors. Our method dynamically selects the next set of measurement beams by maximizing an acquisition function that balances exploration and exploitation over the domain of all possible measurements, improving reconstruction quality while reducing measurement redundancy.

Time series anomaly detection plays a crucial role in a wide range of real-world applications. Reconstruction-based methods have become the mainstream paradigm, but they suffer from over-generalization and under-generalization problems, which are challenging to balance. To address this, we introduce multi-scale clustering to enhance reconstruction-based methods. At the representation level, we integrate the cluster center representations of normal patterns to constrain the model to target representative normal patterns for reconstruction, preventing dominance of powerful capacity and representation capability. At the anomaly criterion level, we derive anomaly confidence score based on cluster membership probability and combine it with reconstruction error, providing dual criteria for detection. Furthermore, the effectiveness of the cluster center representations and anomaly confidence score depends on the clustering performance. Accordingly, we extract neighborhood-centered representations for multi-view clustering to improve clustering performance. Extensive experiments on multiple real-world datasets from diverse application domains demonstrate the state-of-the-art performance of SCAN.

Post-training quantization (PTQ) enables efficient deployment of deep networks using a small set of data. Its application to visual autoregressive models (VAR), however, remains relatively unexplored. We identify two key challenges for applying PTQ to VAR: (i) large reconstruction errors in attention-value products, especially at coarse scales where high attention scores occur more frequently; and (ii) a discrepancy between the sampling frequencies of codebook entries and their predicted probabilities due to limited calibration data. To address these challenges, we propose a PTQ framework tailored for VAR. First, we introduce a shift-and-sum quantization method that reduces reconstruction errors by aggregating quantized results from symmetrically shifted duplicates of value tokens. Second, we present a resampling strategy for calibration data that aligns sampling frequencies of codebook entries with their predicted probabilities. Experiments on class-conditional image generation, inpainting, outpainting, and class-conditional editing show consistent improvements across VAR architectures, establishing a new state of the art in PTQ for VAR.

Physics-informed learning is increasingly used for partial differential equation (PDE)-governed inverse problems, but its reliability remains difficult to certify. This paper develops a no-harm certification-and-selection framework for physics-informed inverse learning. A learned reconstruction is accepted only when its residual-calibrated radius is no worse than the baseline radius, namely when $R_{\mathrm{learn}}\le R_{\mathrm{base}}+\varepsilon_{\mathrm{safe}};$otherwise, the method returns the baseline. The certificate combines data, physics, boundary or initial-condition, and optimization residuals. Under a conditional stability estimate, these residuals yield an a posteriori reconstruction-error bound and a deterministic uncertainty radius. A high-probability certificate is also derived for physics residuals estimated from independent random collocation points. Numerical tests on Poisson source recovery, inverse heat reconstruction, limited-angle tomography, elliptic coefficient identification, and stochastic residual validation show that the selector accepts certified improvements, rejects shifted, hallucinated, or unfinished candidates, and becomes conservative in strongly ill-posed regimes. The framework is therefore a certification-and-selection layer, not another reconstruction architecture.

AI-generated videos are becoming increasingly realistic, raising serious concerns about misinformation, content authenticity, and media trust. Reliable AI-generated video detection is therefore essential for multimedia forensics, yet remains challenging due to the need to capture spatial artifacts, temporal dynamics, and generalize to evolving generative models. In this paper, we explore reconstruction error as a discriminative forensic cue for AI-generated video detection. By reconstructing input videos with a pretrained WF-VAE, we observe that real and generated videos exhibit distinguishable frame-wise reconstruction error patterns, suggesting that reconstruction errors can reveal their distributional discrepancies. However, extending reconstruction-based image detection to videos is non-trivial, since video reconstruction errors are temporally organized across frames and require semantic context for effective interpretation. To address these challenges, we propose ReConFuse, a reconstruction-guided semantic fusion framework for video-level AI-generated video detection. ReConFuse extracts reconstruction error cues from WF-VAE reconstructed videos, aligns them with multi-frame semantic features, and uses a Mamba-based module to model temporal evolution for video-level classification. Experiments across multiple generators and evaluation settings demonstrate the effectiveness and strong generalization ability of ReConFuse.

Recently, sparse autoencoders (SAEs) have emerged as an attractive tool for interpreting and interacting with representations in practical neural networks. While it is common empirical folklore, we also show theoretically that SAEs are highly unstable: different training runs are likely to produce different concept dictionaries and sparse codes. We characterize the model properties that hinder the stability of real-world SAEs, and address each of these problems through minimal changes to the architecture and training procedure. Together, these changes yield two versions of an \textbf{i}dentifiable SAE (iSAE), a variant of the standard TopK SAE with lower reconstruction error and improved stability. We explain this improvement theoretically by connecting SAEs with traditional dictionary learning approaches, and show that the dictionaries learned in practice satisfy an approximate restricted isometry condition, rendering the corresponding sparse codes in those models near-identifiable.

We study the reconstruction of implied volatility surfaces from sparse and noisy option quotes using deep learning models under no-arbitrage constraints. We compare multiple neural architectures, including multilayer perceptrons, convolutional networks, U-Nets, variational autoencoders, and Transformer-based models against classical SVI parameterizations on option market data. Results show that Transformer and U-Net architectures achieve strong reconstruction accuracy, particularly under sparse observation regimes, while soft arbitrage penalties significantly reduce arbitrage violations with moderate impact on reconstruction error. We further analyze the trade-off between accuracy and arbitrage consistency across architectures and regularization strengths.

We study the problem of reconstructing an unknown point in $\mathbb{R}^d$ from approximate linear queries. This setting arises naturally in applications ranging from low-dimensional remote sensing and signal recovery to high-dimensional data analysis and privacy-sensitive inference. Our main goal is to characterize the optimal reconstruction error as a function of the number of queries $T$, the ambient dimension $d$, and the noise parameter $δ$. We first analyze the limit $T \to \infty$ and show that the optimal reconstruction error converges to the explicit value $\sqrt{2d/(d+1)} δ$, which plays a role analogous to the Bayes optimal error in supervised learning. When the dimension is fixed, we show that the excess error above this limit decays doubly exponentially fast as $T \to \infty$, a rate that is significantly faster than those typically encountered in learning curves. When the dimension grows, we show that a number of queries on the order of $\exp(d)$ is necessary and sufficient to achieve vanishing excess error. Finally, we introduce and analyze an improper variant of the reconstruction problem. From a technical perspective, our main contribution is a generalization of Jung's theorem (1901). The classical theorem bounds the maximum possible radius of a set of diameter 1 and characterizes extremal bodies. Our generalization provides a robust variant that characterizes near-extremal bodies and is proved via geometric and dynamical arguments exploiting symmetry and Lie group actions.

Learned image compression has achieved competitive rate-distortion performance, but very-low-bitrate reconstruction remains difficult because the transmitted representation often cannot preserve fine textures and local structures. Perceptual and generative codecs address this problem by using learned reconstruction priors, and controllable codecs allow one model to cover different bitrate and reconstruction preferences. However, controllability alone does not resolve the decoder-side reconstruction-prior problem: under severe bit constraints, the decoder must infer missing details from limited transmitted information, while existing codebook-based controllable designs generally rely on single-codebook token-based priors. This paper proposes Adaptive Fused Prior Transfer for Controllable Generative Image Compression (AFP-GIC), a controllable codec that transfers an adaptive fused prior from a frozen pretrained AdaCode model. Encoder-side fused-prior features guide latent formation, while the decoder predicts a compatible fused prior from the compressed representation and selected control variables, enabling prior-guided reconstruction without transmitting the fused prior itself. A motivating analysis relates decoder-side fused-prior alignment to a reconstruction-error upper bound and shows that the fused-prior family contains single-codebook choices as special cases. Under the unified benchmark, AFP-GIC reduces decoder latency by 18.1% and the overall parameter count by 31.10 million (20.5%) relative to DC-VIC. Experiments on Kodak, CLIC2020, and DIV2K show competitive PSNR, with the clearest perceptual gains in NIQE scores and very-low-bitrate visual comparisons.

Dropout-based sparse-view 3D Gaussian Splatting (3DGS) methods alleviate overfitting by randomly suppressing Gaussian primitives during training. Existing methods mainly focus on designing increasingly sophisticated dropout strategies, while they overlook the resulting inconsistencies among different dropped Gaussian subsets. This oversight often leads to unstable reconstruction and suboptimal Gaussian representation learning.In this paper, we revisit dropout-based sparse-view 3DGS from a consistency regularization perspective and propose PairDropGS, a Paired Dropout-induced Consistency Regularization framework for sparse-view Gaussian splatting. Specifically, PairDropGS first constructs a pair of the dropped Gaussian subsets from a shared Gaussian field and designs a low-frequency consistency regularization to constrain their low-frequency rendered structures. This design encourages the shared Gaussian field to preserve stable scene layout and coarse geometry under different random dropouts, while avoiding excessive constraints on ambiguous high-frequency details. Moreover, we introduce a progressive consistency scheduling strategy to gradually strengthen the consistency regularization during training for stability and robustness of reconstruction. Extensive experiments on widely-used sparse-view benchmarks demonstrate that PairDropGS achieves superior training stability, significantly outperforms existing dropout-based 3DGS methods in reconstruction quality, while exhibiting the simplicity and plug-and-play nature for improving dropout-based optimization.

In this work, we propose Prologue, an approach to bridging the reconstruction-generation gap in autoregressive (AR) image generation. Instead of modifying visual tokens to satisfy both reconstruction and generation, Prologue generates a small set of prologue tokens prepended to the visual token sequence. These prologue tokens are trained exclusively with the AR cross-entropy (CE) loss, while visual tokens remain dedicated to reconstruction. This decoupled design lets us optimize generation through the AR model's true distribution without affecting reconstruction quality, which we further formalize from an ELBO perspective. On ImageNet 256x256, Prologue-Base reduces gFID from 21.01 to 10.75 without classifier-free guidance while keeping reconstruction almost unchanged; Prologue-Large reaches a competitive rFID of 0.99 and gFID of 1.46 using a standard AR model without auxiliary semantic supervision. Interestingly, driven only by AR gradients, prologue tokens exhibit emergent semantic structure: linear probing on 16 prologue tokens reaches 35.88% Top-1, far above the 23.71% of the first 16 tokens from a standard tokenizer; resampling with fixed prologue tokens preserves a similar high-level semantic layout. Our results suggest a new direction: generation quality can be improved by introducing a separate learned generative representation while leaving the original representation intact.

While 3D Gaussian Splatting (3DGS) achieves real-time photorealistic rendering, its performance degrades significantly when training images contain transient objects that violate multi-view consistency. Existing methods face a circular dependency: accurate transient detection requires a well-reconstructed static scene, while clean reconstruction itself depends on reliable transient masks. We address this challenge with DualSplat, a Failure-to-Prior framework that converts first-pass reconstruction failures into explicit priors for a second reconstruction stage. We observe that transients, which appear in only a subset of views, often manifest as incomplete fragments during conservative initial training. We exploit these failures to construct object-level pseudo-masks by combining photometric residuals, feature mismatches, and SAM2 instance boundaries. These pseudo-masks then guide a clean second-pass 3DGS optimization, while a lightweight MLP refines them online by gradually shifting from prior supervision to self-consistency. Experiments on RobustNeRF and NeRF On-the-go show that DualSplat outperforms existing baselines, demonstrating particularly clear advantages in transient-heavy scenes and transient regions.

We consider the problem of reconstructing a HxWx31 hyperspectral image from a $H\times W$ grayscale snapshot measurement that is captured using only a single diffractive lens and a filterless panchromatic photosensor. This problem is severely ill-posed, but we present a model that produces high-quality results in simulation and experiment. We make efficient use of limited training data by creating a conditional denoising diffusion model that operates on small patches in a shift-invariant manner. During inference, we synchronize per-patch hyperspectral predictions using guidance by physical consistency with the system's optical point spread function. Our experiments reveal that the patch size can be as small as the point spread function, with local optical cues being the main source of information about complete spectra. Also, by drawing multiple samples, our model provides per-pixel uncertainty estimates that strongly correlate with reconstruction error.

Feed-forward 3D reconstruction models output a per-pixel confidence that is used by downstream systems as an uncertainty signal. The confidence is trained to serve as a weight in the training loss of models. Whether the confidence can be used as an uncertainty magnitude has not been measured. We audit seven backbones on 13 datasets and score the confidence on four properties, i.e., ranking of error, ratio of error to uncertainty on average, slope of this ratio across the confidence range, and coverage of the implied error distribution. Although the confidence ranks error quite well, the uncertainty decoded from the confidence is too small compared to the actual error. The uncertainty has the right size only under the exact training conditions. The median case is off by at least 2.4x across all seven models, while the uncertainty is further off the more confident the model is. Our work shows that the overconfidence appears on unseen scenes even when the model reaches its loss's optimum. As a post-hoc repair we fit a power law on the confidence with two constants per backbone--dataset pair. The repair brings all four audited properties to target at the dataset level, while leaving ranking untouched. Fitted with the target dataset held out, the constants bring the median case from 2.4x off to 1.35x. The repair does not hold below the dataset level, where two-thirds of held-out scenes are still more than five points off in coverage. We attribute what the repair cannot reach to the model, which carries neither the scale of the error nor the shape of its distribution across predictions. We release the audit protocol, its results, and the fitted constants per backbone-dataset pair.