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.

Ex Input:
It is well-known that deeper neural networks are harder to train than shallower ones. In this short paper, we use the (full) eigenvalue spectrum of the Hessian to explore how the loss landscape changes as the network gets deeper, and as residual connections are added to the architecture. Computing a series of quantitative measures on the Hessian spectrum, we show that the Hessian eigenvalue distribution in deeper networks has substantially heavier tails (equivalently, more outlier eigenvalues), which makes the network harder to optimize with first-order methods. We show that adding residual connections mitigates this effect substantially, suggesting a mechanism by which residual connections improve training.

Ex Output:
Network depth increases outlier eigenvalues in the Hessian. Residual connections mitigate this.


Ex Input:
We characterize the singular values of the linear transformation associated with a standard 2D multi-channel convolutional layer, enabling their efficient computation.   This characterization also leads to an algorithm for projecting a convolutional layer onto an operator-norm ball. We show that this is an effective regularizer;  for example, it improves the test error of a deep residual network using batch normalization on CIFAR-10 from 6.2% to 5.3%.

Ex Output:
We characterize the singular values of the linear transformation associated with a standard 2D multi-channel convolutional layer, enabling their efficient computation. 


Ex Input:
Extreme Classification Methods have become of paramount importance, particularly for Information Retrieval (IR) problems, owing to the development of smart algorithms that are scalable to industry challenges. One of the prime class of models that aim to solve the memory and speed challenge of extreme multi-label learning is Group Testing. Multi-label Group Testing (MLGT) methods construct label groups by grouping original labels either randomly or based on some similarity and then train smaller classifiers to first predict the groups and then recover the original label vectors. Recently, a novel approach called MACH (Merged Average Classifiers via Hashing) was proposed which projects the huge label vectors to a small and manageable count-min sketch (CMS) matrix and then learns to predict this matrix to recover the original prediction probabilities. Thereby, the model memory scales O(logK) for K classes. MACH is a simple algorithm which works exceptionally well in practice. Despite this simplicity of MACH, there is a big gap between the theoretical understanding of the trade-offs with MACH. In this paper we fill this gap. Leveraging the theory of count-min sketch we provide precise quantification of the memory-identifiablity tradeoffs. We extend the theory to the case of multi-label classification, where the dependencies make the estimators hard to calculate in closed forms. To mitigate this issue, we propose novel quadratic approximation using the Inclusion-Exclusion Principle. Our estimator has significantly lower reconstruction error than the typical CMS estimator across various values of number of classes K, label sparsity and compression ratio.

Ex Output:

How to estimate original probability vector for millions of classes from count-min sketch measurements -  a theoretical and practical setup.