Analysis of Basis Pursuit Via Capacity Sets

Finding the sparsest solution $\alpha$ for an under-determined linear system of equations $D\alpha=s$ is of interest in many applications. This problem is known to be NP-hard. Recent work studied conditions on the support size of $\alpha$ that allow its recovery using L1-minimization, via the Basis Pursuit algorithm. These conditions are often relying on a scalar property of $D$ called the mutual-coherence. In this work we introduce an alternative set of features of an arbitrarily given $D$, called the "capacity sets". We show how those could be used to analyze the performance of the basis pursuit, leading to improved bounds and predictions of performance. Both theoretical and numerical methods are presented, all using the capacity values, and shown to lead to improved assessments of the basis pursuit success in finding the sparest solution of $D\alpha=s$

StarNet: towards Weakly Supervised Few-Shot Object Detection

Few-shot detection and classification have advanced significantly in recent years. Yet, detection approaches require strong annotation (bounding boxes) both for pre-training and for adaptation to novel classes, and classification approaches rarely provide localization of objects in the scene. In this paper, we introduce StarNet - a few-shot model featuring an end-to-end differentiable non-parametric star-model detection and classification head. Through this head, the backbone is meta-trained using only image-level labels to produce good features for jointly localizing and classifying previously unseen categories of few-shot test tasks using a star-model that geometrically matches between the query and support images (to find corresponding object instances). Being a few-shot detector, StarNet does not require any bounding box annotations, neither during pre-training nor for novel classes adaptation. It can thus be applied to the previously unexplored and challenging task of Weakly Supervised Few-Shot Object Detection (WS-FSOD), where it attains significant improvements over the baselines. In addition, StarNet shows significant gains on few-shot classification benchmarks that are less cropped around the objects (where object localization is key)

The Journal of Fourier Analysis and Applications Analysis of the Basis Pursuit Via the Capacity Sets

ABSTRACT. Finding the sparsest solution α for an under-determined linear system of equations Dα = s is of interest in many applications. This problem is known to be NP-hard. Recent work studied conditions on the support size of α that allow its recovery using ℓ1-minimization, via the Basis Pursuit algorithm. These conditions are often relying on a scalar property of D called the mutual-coherence. In this work we introduce an alternative set of features of an arbitrarily given D, called the capacity sets. We show how those could be used to analyze the performance of the basis pursuit, leading to improved bounds and predictions of performance. Both theoretical and numerical methods are presented, all using the capacity values, and shown to lead to improved assessments of the basis pursuit success in finding the sparest solution of Dα = s. 1

Learned Shrinkage Approach for Low-Dose Reconstruction in Computed Tomography

We propose a direct nonlinear reconstruction algorithm for Computed Tomography (CT), designed to handle low-dose measurements. It involves the filtered back-projection and adaptive nonlinear filtering in both the projection and the image domains. The filter is an extension of the learned shrinkage method by Hel-Or and Shaked to the case of indirect observations. The shrinkage functions are learned using a training set of reference CT images. The optimization is performed with respect to an error functional in the image domain that combines the mean square error with a gradient-based penalty, promoting image sharpness. Our numerical simulations indicate that the proposed algorithm can manage well with noisy measurements, allowing a dose reduction by a factor of 4, while reducing noise and streak artifacts in the FBP reconstruction, comparable to the performance of a statistically based iterative algorithm