Avi Gupta

Ben Adcock, Avi Gupta

A key problem in approximation theory is the recovery of high-dimensional functions from samples. In many cases, the functions of interest exhibit anisotropic smoothness, and, in many practical settings, the nature of this anisotropy may be unknown a priori. Therefore, an important question involves the development of universal algorithms, namely, algorithms that simultaneously achieve optimal or near-optimal rates of convergence across a range of different anisotropic smoothness classes. In this work, we consider universal approximation of periodic functions that belong to anisotropic Sobolev spaces and anisotropic dominating mixed smoothness Sobolev spaces. Our first result is the construction of a universal algorithm. This recasts function recovery as a sparse recovery problem for Fourier coefficients and then exploits compressed sensing to yield the desired approximation rates. Note that this algorithm is nonadaptive, as it does not seek to learn the anisotropic smoothness of the target function. We then demonstrate optimality of this algorithm up to a dimension-independent polylogarithmic factor. We do this by presenting a lower bound for the adaptive $m$-width for the unit balls of such function classes. Finally, we demonstrate the necessity of nonlinear algorithms. We show that universal linear algorithms can achieve rates that are at best suboptimal by a dimension-dependent polylogarithmic factor. In other words, they suffer from a curse of dimensionality in the rate -- a phenomenon which justifies the necessity of nonlinear algorithms for universal recovery.

Saurabh Yadav, Avi Gupta, Koteswar Rao Jerripothula

The emergence of large foundation models has propelled significant advances in various domains. The Segment Anything Model (SAM), a leading model for image segmentation, exemplifies these advances, outperforming traditional methods. However, such foundation models often suffer from performance degradation when applied to complex tasks for which they are not trained. Existing methods typically employ adapter-based fine-tuning strategies to adapt SAM for tasks and leverage high-frequency features extracted from the Fourier domain. However, Our analysis reveals that these approaches offer limited benefits due to constraints in their feature extraction techniques. To overcome this, we propose \textbf{\textit{SAMwave}}, a novel and interpretable approach that utilizes the wavelet transform to extract richer, multi-scale high-frequency features from input data. Extending this, we introduce complex-valued adapters capable of capturing complex-valued spatial-frequency information via complex wavelet transforms. By adaptively integrating these wavelet coefficients, SAMwave enables SAM's encoder to capture information more relevant for dense prediction. Empirical evaluations on four challenging low-level vision tasks demonstrate that SAMwave significantly outperforms existing adaptation methods. This superior performance is consistent across both the SAM and SAM2 backbones and holds for both real and complex-valued adapter variants, highlighting the efficiency, flexibility, and interpretability of our proposed method for adapting segment anything models.