Interpolation

Community detection is a central problem in graph analysis, with applications ranging from network science to graph signal processing. In recent years, Graph Neural Networks (GNNs) have emerged as effective tools for learning low-dimensional representations of graph-structured data and have shown strong performance in clustering tasks, particularly on large and high-dimensional graphs. This paper investigates the use of GNN-based community detection within a graph signal interpolation framework. After reviewing the main classes of GNN architectures for community detection according to a standard taxonomy, we integrate the resulting graph communities into a Partition of Unity Method (PUM) for interpolation with Graph Basis Functions (GBFs). In this approach, GNN-derived communities are used to construct local subdomains on which GBF interpolants are computed and subsequently combined into a global approximation. Numerical experiments on benchmark %graph datasets, including geometric and urban network examples demonstrate that the proposed combination of GNN-based clustering and GBF-PUM interpolation yields accurate signal reconstructions. The results indicate that deep learning-based community detection can provide effective graph partitions for localized interpolation schemes, supporting its use in scalable graph signal analysis.

Through-plane resolution in clinical MRI is typically much coarser than in-plane resolution, limiting diagnostic utility. This work investigates deep learning approaches to interpolate intermediate MRI slices in prostate imaging, effectively doubling through-plane resolution. I evaluated five architectures (CNN, U-Net, two GAN variants, and DDPM) and discovered that problem formulation has dramatically more impact than architectural complexity. By reformulating the interpolation task to use adjacent slices (i-1, i+1) rather than distant slices (i-2, i+2), I achieved a 58% improvement in SSIM performance across all deterministic architectures. The U-Net model achieved the best results with PSNR of 30.08 dB and SSIM of 0.898, representing a 10.1% improvement over linear interpolation baseline. A DDPM was also evaluated but showed poor reconstruction quality due to fundamental mismatch between stochastic generation and deterministic reconstruction requirements. These findings demonstrate that problem formulation can have 290x more impact than architectural sophistication in medical imaging tasks.

Latent flow matching for image generation usually transports Gaussian noise to variational autoencoder latents along linear paths. Both endpoints, however, concentrate in thin spherical shells, and a Euclidean chord leaves those shells even when preprocessing aligns their radii. By decomposing each latent token into radial and angular components, we show through component-swap probes that decoded perceptual and semantic content is carried predominantly by direction, with radius contributing much less. We therefore project data latents onto a fixed token radius, use the radial projection of Gaussian noise as the spherical prior, finetune the decoder with the encoder frozen, and replace linear interpolation with spherical linear interpolation. The resulting geodesic paths stay on the sphere at every timestep, and their velocity targets are purely angular by construction. Under matched training, the method consistently improves class-conditional ImageNet-256 FID across different image tokenizers, leaves the diffusion architecture unchanged, and requires no auxiliary encoder or representation-alignment objective.

Flow Matching is a powerful framework for learning transport maps between probability distributions. Yet its standard single-parameter formulation is not designed to capture multi-parameter variations where the resulting transport should be path-independent. Path independence is crucial because it ensures that transformations depend only on the initial and target distributions, not on the specific path. In this work, we introduce Path-independent Flow Matching (PiFM), a method for learning vector fields whose induced flows yield path-independent transport between distributions. We show that PiFM generalizes Flow Matching to higher-dimensional parameter domains while enforcing structural conditions that ensure consistency of composed transformations. In addition, we show that, under suitable assumptions, PiFM approximates the Wasserstein barycenter, linking the framework to a notion of distributional interpolation. To enable practical training, we propose a tractable, simulation-free objective that regresses onto multi-parameter conditional probability paths. We showcase empirically that PiFM outperforms other approaches on both synthetic and real world data in interpolating path-independent trajectories and generating desired out of distribution samples.

Stable Diffusion (SD) has evolved DDPM (Denoising Diffusion Probabilistic Model) based image generation significantly by denoising in latent space instead of feature space. This popularized DDPM-based image generation as the cost and compute barrier was significantly lowered. However, these models could only generate fixed-resolution images according to their training configuration. When we attempt to generate higher resolutions, the resulting images show object duplication artifacts consistently. To solve this problem without finetuning SD models, recent works have tried dilating the convolution kernels of the models and have achieved a great level of success. But dilated kernels are harder to fine-tune due to being zero-gapped. Apart from this, other methods, such as patched diffusion, could not solve the object-duplication problem efficiently. Hence, to overcome the limitations of dilated convolutions, we propose kernel interpolation of SD models for higher-resolution image generation. In this work, we show mathematically that interpolation can correctly scale convolution kernels if multiplied by a constant coefficient and achieve competitive empirical results in generating beyond-training-resolution images with Stable Diffusion using zero training. Furthermore, we demonstrate that our method enables interpolation of deep neural networks to adapt to higher-dimensional training data, with a worst-case performance drop of $2.6\%$ in accuracy and F1-Score relative to the baseline. This shows the applicability of our method to be general, where we interpolate fully-connected layers, going beyond convolution layers. We also discuss how we can reduce the memory footprints of training neural networks, using our method up to at least $4\times$.

Neural operators (NOs) are designed to learn maps between infinite-dimensional function spaces. We propose a novel reframing of their use. By introducing an auxiliary base-space, any finite-dimensional function can be viewed as an operator acting by composition on functions of the base-space. Through a range of benchmarks on analytic functions of increasing complexity and dimensionality, we demonstrate that NOs can match or outperform standard multilayer perceptrons and Kolmogorov--Arnold Networks in accuracy while requiring significantly fewer parameters and training time. As a real-world application, we apply a two-dimensional Tensorized Fourier Neural Operator (TFNO) to the nuclear chart, learning a correction to state-of-the-art nuclear mass models as a partially observed residual field. A TFNO ensemble reaches a held-out root-mean-square error of 198.2 keV, placing it among the best recent neural-network approaches while retaining high parameter efficiency and short training times. More broadly, these results introduce NOs as a scalable framework for finite-dimensional function interpolation, from analytic benchmarks to structured scientific data.

Estimating camera geometry typically involves solving minimal problems formulated as systems of multivariate polynomial equations, which often pose computational challenges when using existing Gröbner-basis or resultant-based methods due to matrix inversion needed in the online solver. Here we propose a sampling-based, matrix inversion-free method that constructs the solvers using sparse hidden-variable resultants. The determinant polynomial in the hidden variable is efficiently reconstructed via inverse fast Fourier transform interpolation from sampled evaluations, avoiding symbolic expansion. Solving this polynomial yields the hidden variable, and the remaining unknowns are recovered by identifying rank-1 deficient submatrices and applying Cramer's rule. A greatest common divisor-based criterion ensures robust submatrix identification under noise. Experiments on diverse minimal problems demonstrate that the proposed solver achieves strong numerical stability and competitive runtime, particularly for small-scale problems, providing a practical alternative to traditional Gröbner-basis and resultant-based solvers.

We propose a plug-and-play framework for speech enhancement and separation that augments predictive methods with a generative speech prior. Our approach, termed Stochastic Interpolant Prior for Speech (SIPS), builds on stochastic interpolants and leverages their flexibility to bridge predictive and generative modeling. Specifically, we decompose the interpolation dynamics into a task-specific drift and a stochastic denoising component, allowing a predictive estimate to be integrated directly into the generative sampling process. This results in a mathematically grounded framework for combining strong pretrained predictors with the expressive power of generative models. To this end, we train a score model using only clean speech, yielding a degradation-agnostic prior that can be reused across tasks. During inference, the predictor provides a deterministic drift that steers the sampling process toward a task-consistent estimate, while the score model preserves perceptual naturalness. Unlike prior hybrid approaches, which typically rely on architecture-specific conditioning and are tied to particular predictors or degradation settings, SIPS provides a unified framework that generalizes across predictors and additive degradation tasks. We demonstrate its effectiveness for both speech enhancement and speech separation using recent predictors such as SEMamba and FlexIO. The proposed method consistently improves perceptual quality, achieving gains up +1.0 NISQA for speech separation.

Variational quantum circuits with angle encoding implement truncated Fourier series, and architectures arranging $N$ qubits with $L$ encoding layers each -- sharing encoding budget $E = NL$ -- generate identical frequency spectra, identical frequency redundancy, and require the same minimum parameter count for coefficient control. Despite this equivalence, trainability varies substantially with architecture shape $(N,L)$ at fixed $E$. We identify structural rank deficiency of the coefficient matching Jacobian $J$ as the mechanism responsible. For serial single-qubit architectures, we prove $\mathrm{rank}(J) \leq 2L+1$ regardless of parameter count $P$, with $\dim(\ker J) \geq P-(2L+1)$ growing without bound -- a phenomenon we term \emph{structural gradient starvation}: a growing fraction of parameters become structurally decoupled from the loss as $P$ increases at fixed $L$. Parallel architectures avoid this via independent phase trajectories, ensuring $σ_{\min}(J^{(\mathrm{par})}) > 0$ generically for $P \leq 2E+1$, so no parameter lies in $\ker J$. For practitioners, we further show that the two natural routes to increasing parameter count have fundamentally different effects: adding feature map (FM) layers monotonically strengthens the Jacobian QFIM eigenvalue spectrum and achieves $R^2 \geq 0.95$ with $1.6$--$2.2\times$ fewer parameters than adding trainable blocks across all tested architectures, while trainable blocks improve training only through the classical interpolation mechanism with no quantum-specific benefit.

Flow matching has become a leading framework for generative modeling, but quantifying the uncertainty of its samples remains an open problem. Existing approaches retrain the model with auxiliary variance heads, maintain costly ensembles, or propagate approximate covariance through many integration steps, trading off training cost, inference cost, or accuracy. We show that none of these trade-offs is necessary. By extending Tweedie's formula from the denoising setting to the flow matching interpolant, we derive an exact, closed-form expression for the posterior covariance at every point along the generative trajectory. The result depends on a single quantity, namely the divergence of the learned velocity field, which can be computed post-hoc on any pre-trained flow matching model, requiring no retraining and no architectural modification. For one-step generators such as MeanFlow, the same formula yields the end-to-end generation uncertainty in a single forward pass, eliminating the multi-step variance propagation required by all prior methods. Experiments on MNIST confirm that the resulting per-pixel uncertainty maps are semantically meaningful, concentrating on digit boundaries where inter-sample variation is highest, and that the scalar uncertainty score tracks actual prediction error, all at roughly $10^4 \times$ less total compute than ensembling or Monte Carlo dropout.

Deep learning models are widely deployed in safety-critical domains, but remain vulnerable to adversarial attacks. In this paper, we study the adversarial robustness of NTK neural networks in the context of nonparametric regression. We establish minimax optimal rates for adversarial regression in Sobolev spaces and then show that NTK neural networks, trained via gradient flow with early stopping, can achieve this optimal rate. However, in the overfitting regime, we prove that the minimum norm interpolant is vulnerable to adversarial perturbations.

Facilitating cross-lingual transfer in multilingual language models remains a critical challenge. Towards this goal, we propose an embedding-based data augmentation technique called XITE. We start with unlabeled text from a low-resource target language, identify an English counterpart in a task-specific training corpus using embedding-based similarities and adopt its label. Next, we perform a simple interpolation of the source and target embeddings to create synthetic data for task-specific fine-tuning. Projecting the target text into a language-rich subspace using linear discriminant analysis (LDA), prior to interpolation, further boosts performance. Our cross-lingual embedding-based augmentation technique XITE yields significant improvements of up to 35.91% for sentiment analysis and up to 81.16% for natural language inference, using XLM-R, for a diverse set of target languages including Korean, Arabic, Urdu and Hindi. Apart from boosting cross-lingual transfer, adaptation using XITE also safeguards against forgetting and maintains task performance on the high-resource language.

Exo-to-Ego video generation aims to synthesize a first-person video from a synchronized third-person view and corresponding camera poses. While paired supervision is available, synchronized exo-ego data inherently introduces substantial spatio-temporal and geometric discontinuities, violating the smooth-motion assumptions of standard video generation benchmarks. We identify this synchronization-induced jump as the central challenge and propose Syn2Seq-Forcing, a sequential formulation that interpolates between the source and target videos to form a single continuous signal. By reframing Exo2Ego as sequential signal modeling rather than a conventional condition-output task, our approach enables diffusion-based sequence models, e.g. Diffusion Forcing Transformers (DFoT), to capture coherent transitions across frames more effectively. Empirically, we show that interpolating only the videos, without performing pose interpolation already produces significant improvements, emphasizing that the dominant difficulty arises from spatio-temporal discontinuities. Beyond immediate performance gains, this formulation establishes a general and flexible framework capable of unifying both Exo2Ego and Ego2Exo generation within a single continuous sequence model, providing a principled foundation for future research in cross-view video synthesis.

Dynamic urban environments are often captured by cameras placed at spatially separated locations with little or no view overlap. However, most existing 4D reconstruction methods assume densely overlapping views and struggle under sparse multi-location observations, producing unstable reconstructions in unobserved intermediate regions. To address this practical yet underexplored setting, we propose Stitch4D, a unified 4D reconstruction framework that compensates for missing spatial coverage in sparsely observed urban environments. Stitch4D synthesizes intermediate bridge views between distant camera locations and jointly optimizes real and synthesized observations in a unified coordinate frame with inter-location consistency constraints. By recovering intermediate spatial coverage before optimization, Stitch4D mitigates geometric collapse and improves reconstruction stability in sparse regions. To evaluate this setting, we introduce Urban Sparse 4D (U-S4D), a controlled CARLA-based benchmark for free-viewpoint reconstruction under sparse multi-location configurations. Experiments on U-S4D show that Stitch4D consistently outperforms representative 4D reconstruction baselines in image-quality metrics. These results suggest that recovering intermediate spatial coverage is an effective strategy for stabilizing 4D reconstruction in sparse urban environments. The project page is provided in https://stitch4d-project-page.vercel.app/.

We develop a kernel method for generative modeling within the stochastic interpolant framework, replacing neural network training with linear systems. The drift of the generative SDE is $\hat b_t(x) = \nablaφ(x)^\topη_t$, where $η_t \in \mathbb{R}^P$ solves a $P\times P$ system computable from data, with $P$ independent of the data dimension $d$. Since estimates are inexact, the diffusion coefficient $D_t$ affects sample quality; the optimal $D_t^*$ from Girsanov diverges at $t=0$, but this poses no difficulty and we develop an integrator that handles it seamlessly. The framework accommodates diverse feature maps: scattering transforms, pretrained generative models, etc, enabling generation and model combination without neural network training. We demonstrate the approach on financial time series, turbulence, and image generation.

This paper studies the training-testing discrepancy (a.k.a. exposure bias) problem for improving the diffusion models. During training, the input of a prediction network at one training timestep is the corresponding ground-truth noisy data that is an interpolation of the noise and the data, and during testing, the input is the generated noisy data. We present a novel training approach, named MixFlow, for improving the performance. Our approach is motivated by the Slow Flow phenomenon: the ground-truth interpolation that is the nearest to the generated noisy data at a given sampling timestep is observed to correspond to a higher-noise timestep (termed slowed timestep), i.e., the corresponding ground-truth timestep is slower than the sampling timestep. MixFlow leverages the interpolations at the slowed timesteps, named slowed interpolation mixture, for post-training the prediction network for each training timestep. Experiments over class-conditional image generation (including SiT, REPA, and RAE) and text-to-image generation validate the effectiveness of our approach. Our approach MixFlow over the RAE models achieve strong generation results on ImageNet: 1.43 FID (without guidance) and 1.10 (with guidance) at 256 x 256, and 1.55 FID (without guidance) and 1.10 (with guidance) at 512 x 512.

Learning the dynamics of a process given sampled observations at several time points is an important but difficult task in many scientific applications. When no ground-truth trajectories are available, but one has only snapshots of data taken at discrete time steps, the problem of modelling the dynamics, and thus inferring the underlying trajectories, can be solved by multi-marginal generalisations of flow matching algorithms. This paper proposes a novel flow matching method that overcomes the limitations of existing multi-marginal trajectory inference algorithms. Our proposed method, ALI-CFM, uses a GAN-inspired adversarial loss to fit neurally parametrised interpolant curves between source and target points such that the marginal distributions at intermediate time points are close to the observed distributions. The resulting interpolants are smooth trajectories that, as we show, are unique under mild assumptions. These interpolants are subsequently marginalised by a flow matching algorithm, yielding a trained vector field for the underlying dynamics. We showcase the versatility and scalability of our method by outperforming the existing baselines on spatial transcriptomics and cell tracking datasets, while performing on par with them on single-cell trajectory prediction. Code: https://github.com/mmacosha/adversarially-learned-interpolants.

Prevailing activation functions such as Swish and GELU tend toward domain-specific optima, Swish was discovered via neural architecture search on vision benchmarks, while GELU dominates transformer-based language models, and neither offers any mechanism to adapt its gating shape to individual layers. This rigidity is especially consequential in transformer FFN blocks, where LayerNorm, unlike BatchNorm, does not suppress the gradient pathologies that activation choice induces across depth. We propose SG-Blend, a per layer adaptive activation that combines SSwish, a bias-corrected, parametric Swish variant we also introduce, with learnable sharpness \b{eta} and zero-centering bias γ, with GELU through a per-layer blend coefficient α, letting each layer locate its own optimum along the SSwishGELU continuum at a cost of only three additional scalars per FFN block, with \b{eta} initialized to 1.0 and learned freely via backpropagation. On BERT-style IMDB classification (5 seeds), it matches peak accuracy (81.31%) while reducing seed-to-seed variance by 42% relative to GELU. Furthermore, it generalizes to autoregressive pretraining, achieving the lowest validation perplexity (49.10) on WikiText103 among all baselines. Crucially, ablations confirm the interpolation structure itself drives these gains, delivering reliable, top-tier performance. Beyond natural language processing, we demonstrate that SG-Blend generalizes robustly to a wider variety of tasks, extending its efficacy to computer vision and other diverse domains.

This paper establishes the generalization error of pooled min-$\ell_2$-norm interpolation in transfer learning, where data from diverse distributions are available. Min-norm interpolators arise naturally as implicit regularized limits of modern machine learning algorithms. Prior work has characterized their out-of-distribution risk when samples from the test distribution are unavailable during training. In many applications, however, limited test samples may be available at training time, yet properties of min-norm interpolation in this regime remain poorly understood. We address this gap by characterizing the bias and variance of pooled min-$\ell_2$-norm interpolation under both covariate shift and model shift. Our results yield several important implications. In certain cases under model shift, we show that adding data always hurts when the signal-to-noise ratio (SNR) is low. At higher SNR levels, transfer learning is beneficial provided the shift-to-signal ratio falls below a threshold that we characterize explicitly. Under covariate shift, we find that when the source sample size is small relative to the dimension, greater heterogeneity between domains reduces risk, and vice versa. While our model shift results are initially established for Gaussian designs, we extend them to more general designs through a universality argument. To illustrate the broader applicability of our technical tools beyond interpolation learning, we characterize the risk of a bias-corrected estimator that uses the pooled interpolator as an initialization and corrects the resulting bias with target data. On the technical side, we develop a novel anisotropic local law and a Lindeberg-swapping argument, yielding tools that may be of independent interest in random matrix theory and universality analysis. Finally, we supplement our theory with simulations demonstrating the finite-sample efficacy of our results.

We study continual pre-training (CPT) as a mechanism for adapting general-purpose large language models to specialized domains: mathematics, instruction, code, and natural text. Using singular value decomposition of weight matrices, we find that CPT leaves singular value spectra largely invariant, with adaptation driven mainly by changes in singular vectors. An analysis of attention-head projection matrices reveals strong, domain-dependent head heterogeneity, which we exploit to define a head importance criterion: up to 60% of head updates can be removed without measurable quality loss. Selectively rewinding low-importance heads to their pre-trained state improves benchmark accuracy by up to 4% versus the fully trained baseline. Finally, we identify domain connectivity - linear interpolation between CPT checkpoints yields smooth domain-quality interpolation without notable degradation on either domain - and release Diffract, an open-source toolkit for scalable spectral analysis of billion-parameter models.