Variational representations of $f$-divergences are central to many machine learning algorithms, with Lipschitz constrained variants recently gaining attention. Inspired by this, we define the Moreau-Yosida approximation of $f$-divergences with respect to the Wasserstein-$1$ metric. The corresponding variational formulas provide a generalization of a number of recent results, novel special cases of interest and a relaxation of the hard Lipschitz constraint. Additionally, we prove that the so-called tight variational representation of $f$-divergences can be to be taken over the quotient space of Lipschitz functions, and give a characterization of functions achieving the supremum in the variational representation.

We present a new approach to graph limit theory that unifies and generalizes the two most well-developed directions, namely dense graph limits (even the more general Lp limits) and Benjamini–Schramm limits (even in the stronger local-global setting). We illustrate by examples that this new framework provides a rich limit theory with natural limit objects for graphs of intermediate density. Moreover, it provides a limit theory for bounded operators (called P-operators) of the form L∞(Ω)→L1(Ω) for probability spaces Ω .

Artificial intelligence (AI) has long been one of the big promises of computer science, and related research has fundamentally shaped our thinking about the human-machine relationship. For decades, AI has failed to fulfil its promise of solving practical challenges. This is changing now. As the past ten years have given rise to radical changes and major breakthroughs making AI the major solution in many situations. Machine learning is a set of mathematical techniques which powers most current-generation artificial intelligence systems, notably self-driving cars, intelligent assistants, speech recognition and machine translation software.

A sík legfeljebb mekkora hányada színezhető ki úgy, hogy két kiszínezett pont nem lehet pontosan egység távolságra egymástól? Ezt a geometriai kérdést Leo Moser fogalmazta meg az 1960-as évek elején, Hadwiger és Nelson egy rokon problémájával kapcsolatban. Leo Moser és Erdős Pál sejtése szerint ez a hányad nem érheti el az $\frac{1}{4}$-et; a jelenleg ismert legerősebb, 0,2293 értékű alsó korlátot Hallard Croft 1967-es konstrukciója adja. A problémával kapcsolatban számos kutatócsoport publikált már részeredményeket, amelyek a kezdeti 0,2857-es felső sűrűség-becslést az elmúlt 60 évben fokozatosan 0.