SVT incorporates the scattering network utilizing the DTCWT for image decomposition into low and high-frequency components. Our primary focus is to analyze the low-frequency and high-frequency filter components to emphasize SVT's exceptional directional orientation capabilities.

Throughout the paper, we will use standard notations for probability unless otherwise stated, e.g.,Pr[ ]for probability,p[ ]for density,E[ ]for expectation.

Existing solutions often resort to down-sampling operations, such as pooling, to reduce computational cost. Unfortunately, such operations are non-invertible and can result in information loss.

RDP inherits and generalizes the information-theoretic properties of DP . This composition rule, together with Lemma 3, often allows for tighter calculations of (null,δ)-DP for the composed mechanism than directly invoking the strong composition theorem below. Also w.l.o.g., we assume thresholds Substituting the above expression to the definition of RDP and apply Jensen's inequality (6) = The inequality applies Jensen's inequality to bivariate function We use a trick due to [Bun and Steinke, 2016] with some modifications. Now we are ready to prove the three claims of Theorem 8. 13 The claim (3): Substitute the the above bound into Lemma 17, we get: E In the last line, we applied the "indistinguishability" property of an RDP mechanism in Lemma 15 The issue is how to proceed. The proof follows a similar sequence of arguments to that we presented for c = 1 .

The Sparse V ector Technique (SVT) [Dwork et al., 2009] is a fundamental tool in differential privacy

This paper addresses the problem of visual target tracking in scenarios where a pursuer may experience intermittent loss of visibility of the target. The design of a Switched Visual Tracker (SVT) is presented which aims to meet the competing requirements of maintaining both proximity and visibility. SVT alternates between a visual tracking mode for following the target, and a recovery mode for regaining visual contact when the target falls out of sight. We establish the stability of SVT by extending the average dwell time theorem from switched systems theory, which may be of independent interest. Our implementation of SVT on an Agilicious drone [1] illustrates its effectiveness on tracking various target trajectories: it reduces the average tracking error by up to 45% and significantly improves visibility duration compared to a baseline algorithm. The results show that our approach effectively handles intermittent vision loss, offering enhanced robustness and adaptability for real-world autonomous missions. Additionally, we demonstrate how the stability analysis provides valuable guidance for selecting parameters, such as tracking speed and recovery distance, to optimize the SVT's performance.

Minimizing the rank of a matrix subject to affine constraints is a fundamental problem with many important applications in machine learning and statistics. In this paper we propose a simple and fast algorithm SVP (Singular Value Projection) for rank minimization under affine constraints (ARMP) and show that SVP recovers the minimum rank solution for affine constraints that satisfy a restricted isometry property (RIP). Our method guarantees geometric convergence rate even in the presence of noise and requires strictly weaker assumptions on the RIP constants than the existing methods. We also introduce a Newton-step for our SVP framework to speed-up the convergence with substantial empirical gains. Next, we address a practically important application of ARMP - the problem of lowrank matrix completion, for which the defining affine constraints do not directly obey RIP, hence the guarantees of SVP do not hold. However, we provide partial progress towards a proof of exact recovery for our algorithm by showing a more restricted isometry property and observe empirically that our algorithm recovers low-rank incoherent matrices from an almost optimal number of uniformly sampled entries. We also demonstrate empirically that our algorithms outperform existing methods, such as those of [5, 18, 14], for ARMP and the matrix completion problem by an order of magnitude and are also more robust to noise and sampling schemes. In particular, results show that our SVP-Newton method is significantly robust to noise and performs impressively on a more realistic power-law sampling scheme for the matrix completion problem.

Low-rank matrix completion concerns the problem of estimating unobserved entries in a matrix using a sparse set of observed entries. We consider the non-uniform setting where the observed entries are sampled with highly varying probabilities, potentially with different asymptotic scalings. We show that under structured sampling probabilities, it is often better and sometimes optimal to run estimation algorithms on a smaller submatrix rather than the entire matrix. In particular, we prove error upper bounds customized to each entry, which match the minimax lower bounds under certain conditions. Our bounds characterize the hardness of estimating each entry as a function of the localized sampling probabilities. We provide numerical experiments that confirm our theoretical findings.