We present RELOOP, a domain-specific language for relational optimization embedded in Python. It allows the user to express relational optimization problems in a natural syntax that follows logic and linear algebra, rather than in the restrictive standard form required by solvers, and can automatically compile the model to a lower-order but equivalent model. Moreover, RELOOP makes it easy to combine relational optimization with high-level features of Python such as loops, parallelism and interfaces to relational databases.

I'm definitely interested in this question too, as someone who is self-taught about machine learning and skipped over a lot of the mathematical theory. I know that certain areas are very important, such as matrix factorisation methods which are huge. And algorithms such as neural networks rely heavily on lots of aspects of linear algebra (see https://www.utdallas.edu/ The kernel trick in SVM is also founded in linear algebra (the dot product). It'd be interesting to hear an expert elaborate on this.

Asymptotic factorizations for the small-ball probability (SmBP) of a Hilbert valued random element $X$ are rigorously established and discussed. In particular, given the first $d$ principal components (PCs) and as the radius $\varepsilon$ of the ball tends to zero, the SmBP is asymptotically proportional to (a) the joint density of the first $d$ PCs, (b) the volume of the $d$-dimensional ball with radius $\varepsilon$, and (c) a correction factor weighting the use of a truncated version of the process expansion. Moreover, under suitable assumptions on the spectrum of the covariance operator of $X$ and as $d$ diverges to infinity when $\varepsilon$ vanishes, some simplifications occur. In particular, the SmBP factorizes asymptotically as the product of the joint density of the first $d$ PCs and a pure volume parameter. All the provided factorizations allow to define a surrogate intensity of the SmBP that, in some cases, leads to a genuine intensity. To operationalize the stated results, a non-parametric estimator for the surrogate intensity is introduced and it is proved that the use of estimated PCs, instead of the true ones, does not affect the rate of convergence. Finally, as an illustration, simulations in controlled frameworks are provided.

Julia is a high-level dynamic programming language designed to address the requirements of high-performance numerical and scientific computing while also being effective for general purpose programming.[1][2][3][4] Unusual aspects of Julia's design include having a type system with parametric types in a fully dynamic programming language and adopting multiple dispatch as its core programming paradigm. It allows for parallel and distributed computing; and direct calling of C and Fortran libraries without a compiler without glue code and includes best-of-breed libraries for floating-point, linear algebra, random number generation, fast Fourier transforms, and regular expression matching. Julia's core is implemented in C and C, its parser in Scheme, and the LLVM compiler framework is used for just-in-time generation of machine code. The standard library is implemented in Julia itself, using the Node.js's

A computer algebra system (CAS) is a software program that allows computation over mathematical expressions in a way which is similar to the traditional manual computations of mathematicians and scientists. The development of the computer algebra systems in the second half of the 20th century is part of the discipline of "computer algebra" or "symbolic computation", which has spurred work in algorithms over mathematical objects such as polynomials. Computer algebra systems may be divided in two classes: the specialized ones and the general purpose ones. The specialized ones are devoted to a specific part of mathematics, such as number theory, group theory, or teaching of elementary mathematics. General purpose computer algebra systems aim to be useful to a user working in any scientific field that requires manipulation of mathematical expressions.

Analytics is becoming critical in all part of our lives. Biostatistics has been a big driver of this analytics demand in the field of pharmaceuticals, biotech, health & medicine. Biostatistics (or biometry) is the application of statistics to a wide range of topics in biology. The science of biostatistics encompasses the design of biological experiments, especially in medicine, pharmacy, agriculture and fishery; the collection, summarization, and analysis of data from those experiments; and the interpretation of, and inference from, the results. A major branch of this is medical biostatistics,[1] which is exclusively concerned with medicine and health.

What I found useful during my PhD (this could apply to master program too) is that I immediately started to work for a company on GIS, digital cartography, and water management (predicting extreme floods locally - how much the water could rise, at worse in 100 years, at any (x,y) coordinate on a digital map, modeling how any drop of water falling somewhere runs down, goes underground, eventually reaches low elevation and merges with other water drops on the way down - the digital maps had elevation and land use data available for each pixel; by land use I mean crop, forest, water, rock and so on, as this is important to model how water moves). Very applied and interesting stuff. My first paper (after an article about flood predictions, in a local specialized journal) was in Journal of Number Theory though I never attended classes on number theory. I then started to publish in computational statistics journal, but also in IEEE Pattern Analysis and Machine Intelligence, and Journal of the Royal Statistical Society, series B. I'm currently finishing a book on data science (Wiley, exp. The take away from this is that it helps getting polyvalent, if the PhD/Master student can do applied work for a real company, hired and paid as a real employee (partnership between university and private sector), at the beginning of his program.

A known failing of many popular random graph models is that the Aldous-Hoover Theorem guarantees these graphs are dense with probability one; that is, the number of edges grows quadratically with the number of nodes. This behavior is considered unrealistic in observed graphs. We define a notion of edge exchangeability for random graphs in contrast to the established notion of infinite exchangeability for random graphs --- which has traditionally relied on exchangeability of nodes (rather than edges) in a graph. We show that, unlike node exchangeability, edge exchangeability encompasses models that are known to provide a projective sequence of random graphs that circumvent the Aldous-Hoover Theorem and exhibit sparsity, i.e., sub-quadratic growth of the number of edges with the number of nodes. We show how edge-exchangeability of graphs relates naturally to existing notions of exchangeability from clustering (a.k.a. partitions) and other familiar combinatorial structures.

Crowdsourcing platforms are now extensively used for conducting subjective pairwise comparison studies. In this setting, a pairwise comparison dataset is typically gathered via random sampling, either \emph{with} or \emph{without} replacement. In this paper, we use tools from random graph theory to analyze these two random sampling methods for the HodgeRank estimator. Using the Fiedler value of the graph as a measurement for estimator stability (informativeness), we provide a new estimate of the Fiedler value for these two random graph models. In the asymptotic limit as the number of vertices tends to infinity, we prove the validity of the estimate. Based on our findings, for a small number of items to be compared, we recommend a two-stage sampling strategy where a greedy sampling method is used initially and random sampling \emph{without} replacement is used in the second stage. When a large number of items is to be compared, we recommend random sampling with replacement as this is computationally inexpensive and trivially parallelizable. Experiments on synthetic and real-world datasets support our analysis.

We study the task of learning from non-i.i.d. data. In particular, we aim at learning predictors that minimize the conditional risk for a stochastic process, i.e. the expected loss of the predictor on the next point conditioned on the set of training samples observed so far. For non-i.i.d. data, the training set contains information about the upcoming samples, so learning with respect to the conditional distribution can be expected to yield better predictors than one obtains from the classical setting of minimizing the marginal risk. Our main contribution is a practical estimator for the conditional risk based on the theory of non-parametric time-series prediction, and a finite sample concentration bound that establishes uniform convergence of the estimator to the true conditional risk under certain regularity assumptions on the process.