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.

Example Input: We study the problem of learning and optimizing through physical simulations via differentiable programming. We present DiffSim, a new differentiable programming language tailored for building high-performance differentiable physical simulations. We demonstrate the performance and productivity of our language in gradient-based learning and optimization tasks on 10 different physical simulators. For example, a differentiable elastic object simulator written in our language is 4.6x faster than the hand-engineered CUDA version yet runs as fast, and is 188x faster than TensorFlow. Using our differentiable programs, neural network controllers are typically optimized within only tens of iterations. Finally, we share the lessons learned from our experience developing these simulators, that is, differentiating physical simulators does not always yield useful gradients of the physical system being simulated. We systematically study the underlying reasons and propose solutions to improve gradient quality.
Example Output: We study the problem of learning and optimizing through physical simulations via differentiable programming, using our proposed DiffSim programming language and compiler.

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

Example Input: Deep neural networks are surprisingly efficient at solving practical tasks,
 but the theory behind this phenomenon is only starting to catch up with
 the practice. Numerous works show that depth is the key to this efficiency.
 A certain class of deep convolutional networks – namely those that correspond
 to the Hierarchical Tucker (HT) tensor decomposition – has been
 proven to have exponentially higher expressive power than shallow networks.
 I.e. a shallow network of exponential width is required to realize
 the same score function as computed by the deep architecture. In this paper,
 we prove the expressive power theorem (an exponential lower bound on
 the width of the equivalent shallow network) for a class of recurrent neural
 networks – ones that correspond to the Tensor Train (TT) decomposition.
 This means that even processing an image patch by patch with an RNN
 can be exponentially more efficient than a (shallow) convolutional network
 with one hidden layer. Using theoretical results on the relation between
 the tensor decompositions we compare expressive powers of the HT- and
 TT-Networks. We also implement the recurrent TT-Networks and provide
 numerical evidence of their expressivity.
Example Output:
We prove the exponential efficiency of recurrent-type neural networks over shallow networks.