In this task, you are given the abstract of a research paper. Your task is to generate a summary of this abstract. Your summary should not be very short, but it's better if it's not more than 30 words.

Input: Consider Input: We extend the recent results of (Arora et al., 2019) by a spectral analysis of representations corresponding to kernel and neural embeddings. They showed that in a simple single layer network, the alignment of the labels to the eigenvectors of the corresponding Gram matrix determines both the convergence of the optimization during training as well as the generalization properties. We generalize their result to kernel and neural representations and show that these extensions improve both optimization and generalization of the basic setup studied in (Arora et al., 2019).

Output: Spectral analysis for understanding how different representations can improve optimization and generalization.


Input: Consider Input: Oversubscription planning (OSP) is the problem of finding plans that maximize the utility value of their end state while staying within a specified cost bound. Recently, it has been shown that OSP problems can be reformulated as classical planning problems with multiple cost functions but no utilities.   Here we take advantage of this reformulation to show that OSP problems can be solved optimally using the A* search algorithm, in contrast to previous approaches that have used variations on branch-and-bound search. This allows many powerful techniques developed for classical planning to be applied to OSP problems. We also introduce novel bound-sensitive heuristics, which are able to reason about the primary cost of a solution while taking into account secondary cost functions and bounds, to provide superior guidance compared to heuristics that do not take these bounds into account. We implement two such bound-sensitive variants of existing classical planning heuristics, and show experimentally that the resulting search is significantly more informed than comparable heuristics that do not consider bounds.

Output: We show that oversubscription planning tasks can be solved using A* and introduce novel bound-sensitive heuristics for oversubscription planning tasks.


Input: Consider Input: Memorization of data in deep neural networks has become a subject of significant research interest. 
 In this paper, we link memorization of  images in deep convolutional autoencoders to downsampling through strided convolution.   To analyze this mechanism in a simpler setting, we train linear convolutional autoencoders and show that linear combinations of training data are stored as eigenvectors in the linear operator corresponding to the network when downsampling is used.   On the other hand, networks without downsampling do not memorize training data.   We provide further evidence that the same effect happens in nonlinear networks.   Moreover, downsampling in nonlinear networks causes the model to not only memorize just linear combinations of images, but individual training images.   Since convolutional autoencoder components are building blocks of deep convolutional networks, we envision that our findings will shed light on the important phenomenon of memorization in over-parameterized deep networks.  



Output: We identify downsampling as a mechansim for memorization in convolutional autoencoders.