**Selective Forgetting**

Few-shot learning is a challenging problem where the system is required to achieve generalization from only few examples. Meta-learning tackles the problem by learning prior knowledge shared across a distribution of tasks, which is then used to quickly adapt to unseen tasks. Model-agnostic meta-learning (MAML) algorithm formulates prior knowledge as a common initialization across tasks. However, forcibly sharing an initialization brings about conflicts between tasks and thus compromises the quality of the initialization. In this work, by observing that the extent of compromise differs among tasks and between layers of a neural network, we propose a new initialization idea that employs task-dependent layer-wise attenuation, which we call selective forgetting. The proposed attenuation scheme dynamically controls how much of prior knowledge each layer will exploit for a given task. The experimental results demonstrate that the proposed method mitigates the conflicts and provides outstanding performance as a result. We further show that the proposed method, named L2F, can be applied and improve other state-of-the-art MAML-based frameworks, illustrating its generalizability. … **Uncertainty-Aware Principal Component Analysis**

We present a technique to perform dimensionality reduction on data that is subject to uncertainty. Our method is a generalization of traditional principal component analysis (PCA) to multivariate probability distributions. In comparison to non-linear methods, linear dimensionality reduction techniques have the advantage that the characteristics of such probability distributions remain intact after projection. We derive a representation of the covariance matrix that respects potential uncertainty in each of the observations, building the mathematical foundation of our new method uncertainty-aware PCA. In addition to the accuracy and performance gained by our approach over sampling-based strategies, our formulation allows us to perform sensitivity analysis with regard to the uncertainty in the data. For this, we propose factor traces as a novel visualization that enables us to better understand the influence of uncertainty on the chosen principal components. We provide multiple examples of our technique using real-world datasets and show how to propagate multivariate normal distributions through PCA in closed-form. Furthermore, we discuss extensions and limitations of our approach. … **Linear k-Junta**

We study the problem of testing if a function depends on a small number of linear directions of its input data. We call a function $f$ a \emph{linear $k$-junta} if it is completely determined by some $k$-dimensional subspace of the input space. In this paper, we study the problem of testing whether a given $n$ variable function $f : \mathbb{R}^n \to $, is a linear $k$-junta or $\epsilon$-far from all linear $k$-juntas, where the closeness is measured with respect to the Gaussian measure on $\mathbb{R}^n$. This problems is a common generalization of (i) The combinatorial problem of junta testing on the hypercube which tests whether a Boolean function is dependent on at most $k$ of its variables and (ii) Geometric testing problems such as testing if a function is an intersection of $k$ halfspaces. We prove the existence of a $\mathsf{poly}(k \cdot s/\epsilon)$-query non-adaptive tester for linear $k$-juntas with surface area at most $s$. The polynomial dependence on $s$ is necessary as we provide a $\mathsf{poly}(s)$ lower bound on the query complexity of any non-adaptive test for linear juntas. Moreover, we show that if the function is a linear $k$-junta with surface area at most $s$, then there is a $(s \cdot k)^{O(k)}$-query non-adaptive algorithm to learn the function \emph{up to a rotation of the basis}. {We also use this procedure to obtain a non-adaptive tester (with the same query complexity) for subclasses of linear $k$-juntas closed under rotation.} … **Accelerated Alternating Projection**

We study robust PCA for the fully observed setting, which is about separating a low rank matrix \BL \BL and a sparse matrix \BS \BS from their sum \BD=\BL+\BS \BD=\BL+\BS . In this paper, a new algorithm, dubbed accelerated alternating projections, is introduced for robust PCA which significantly improves the computational efficiency of the existing alternating projections proposed in (Netrapalli et al., 2014) when updating the low rank factor. The acceleration is achieved by first projecting a matrix onto some low dimensional subspace before obtaining a new estimate of the low rank matrix via truncated SVD. Exact recovery guarantee has been established which shows linear convergence of the proposed algorithm. Empirical performance evaluations establish the advantage of our algorithm over other state-of-the-art algorithms for robust PCA. …

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**17**
*Wednesday*
May 2023

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