A systems programmer who uses a statically typed functional language like OCaml, Scala, or Haskell can be forgiven for doubting the importance of the latest addition to the type system of their language of choice.  Certainly, that was what we thought about the arrival of Generalized Algebraic Data Types, or GADTs, to OCaml.  The motivating examples for GADTs all seem to be about tagless interpreters and typeful representations of ADTs, all of which sound like the kind of thing you get when you let compiler writers design your language.

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We consider relative error low rank approximation of tensors with respect to the Frobenius norm: given an order-q tensor A, output a rank-k tensor B for which |A-B|_F≤(1+ϵ)OPT, where OPT =inf_{rank-k A′} |A−A′|_F. Despite the success on obtaining relative error low rank approximations for matrices, no such results were known for tensors. One structural issue is there may be no rank-k tensor A' achieving the above infinum.

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In modern cloud infrastructures, each physical server often runs multiple virtual machines and employs a software Virtual Switch (VS) to handle their traffic. In addition to switching, the VS performs network measurements, such as identifying the most frequent flows, which are essential for networking applications such as load balancing and intrusion detection.

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Combinatorial optimization  captures many natural problems such as matching, load balancing, social welfare, orienteering, network design, clustering, and submodular optimization. Classically, these problems have been studied in the full-information setting, i.e., where the entire input---an objective function with some constraints---is given and the goal is to select a feasible set to maximize/minimize the objective function. In this thesis we focus on combinatorial  problems in an uncertain environment where we only have partial knowledge about the input.

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The rapid progress in artificial intelligence in recent years can largely be attributed to the resurgence of neural networks, which enables learning representations in an end-to-end manner. Although neural network are powerful, they have many limitations. For examples, neural networks are computationally expensive and memory inefficient; Neural network training needs many labeled exampled, especially for tasks that require reasoning and external knowledge.

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Information-flow security-typed programming languages use types to statically enforce noninterference between potentially conspiring values, such as the arguments and results of functions.  But to adopt static security types, like other advanced type disciplines, programmers must adopt the discipline wholesale.

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