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Epsilon (ˈɛpsᵻlɒn, UKalsoɛpˈsaɪlən; uppercase Ε, lowercase ε or lunate ϵ; έψιλον) is the fifth letter of the Greek alphabet, corresponding phonetically to a mid front unrounded vowel e̞ or ɛ̝. In the system of Greek numerals it also has the value five. It was derived from the Phoenician letter He . Letters that arose from epsilon include the Roman E, Ë and Ɛ, and Cyrillic Е, È, Ё, Є and Э. The name of the letter was originally εἶ (êː), but it was later changed to ἒ ψιλόν (e psilon 'simple e') in the Middle Ages to distinguish the letter from the digraph αι, a former diphthong that had come to be pronounced the same as epsilon. The uppercase form of epsilon is identical to Latin E but has its own code point in Unicode: . The lowercase version has two typographical variants, both inherited from medieval Greek handwriting. One, the most common in modern typography and inherited from medieval minuscule, looks like a reversed number "3" and is encoded . The other, also known as lunate

Bounds on multiscalar CFTs in the epsilon expansion

Matthijs Jan Hogervorst

We study fixed points with N scalar fields in 4 - epsilon dimensions to leading order in epsilon using a bottom-up approach. We do so by analyzing O(N) invariants of the quartic coupling lambda(ijkl) that describes such CFTs. In particular, we show that lambda(iijj) and lambda ijkl2 are restricted to a specific domain, refining a result by Rychkov and Stergiou. We also study averages of one-loop anomalous dimensions of composite operators without gradients. In many cases, we are able to show that the O(N) fixed point maximizes such averages. In the final part of this work, we generalize our results to theories with N complex scalars and to bosonic QED. In particular we show that to leading order in epsilon, there are no bosonic QED fixed points with N < 183 flavors.

SPRINGER2021

epsilon-Coresets for Clustering (with Outliers) in Doubling Metrics

Lingxiao Huang, Jian Li

We study the problem of constructing epsilon-coresets for the (k, z)-clustering problem in a doubling metric M(X, d). An epsilon-coreset is a weighted subset S subset of X with weight function w : S -> R->= 0, such that for any k-subset C is an element of X, it holds that Sigma(x is an element of S) w(x).d(z) (x, C) is an element of (1 +/- epsilon) . Sigma(x is an element of X) d(z) (x, C). We present an efficient algorithm that constructs an epsilon-coreset for the (k, z)-clustering problem in M(X, d), where the size of the coreset only depends on the parameters k, z, epsilon and the doubling dimension ddim(M). To the best of our knowledge, this is the first efficient epsilon-coreset construction of size independent of vertical bar X vertical bar for general clustering problems in doubling metrics. To this end, we establish the first relation between the doubling dimension of M(X, d) and the shattering dimension (or VC-dimension) of the range space induced by the distance d. Such a relation is not known before, since one can easily construct instances in which neither one can be bounded by (some function of) the other. Surprisingly, we show that if we allow a small (1 +/- epsilon)-distortion of the distance function d (the distorted distance is called the smoothed distance function), the shattering dimension can be upper bounded by O(epsilon(-O(ddim(M)))). For the purpose of coreset construction, the above bound does not suffice as it only works for unweighted spaces. Therefore, we introduce the notion of tau-error probabilistic shattering dimension, and prove a (drastically better) upper bound of O(ddim(M).log(1/epsilon)+log log 1/tau) for the probabilistic shattering dimension for weighted doubling metrics. As it turns out, an upper bound for the probabilistic shattering dimension is enough for constructing a small coreset. We believe the new relation between doubling and shattering dimensions is of independent interest and may find other applications. Furthermore, we study robust coresets for (k, z)-clustering with outliers in a doubling metric. We show an improved connection between alpha-approximation and robust coresets. This also leads to improvement upon the previous best known bound of the size of robust coreset for Euclidean space [Feldman and Langberg, STOC 11]. The new bound entails a few new results in clustering and property testing. As another application, we show constant-sized (epsilon, k, z)-centroid sets in doubling metrics can be constructed by extending our coreset construction. Prior to our result, constant-sized centroid sets for general clustering problems were only known for Euclidean spaces. We can apply our centroid set to accelerate the local search algorithm (studied in [Friggstad et al., FOCS 2016]) for the (k, z)-clustering problem in doubling metrics.

IEEE COMPUTER SOC2018

epsilon-Coresets for Clustering (with Outliers) in Doubling Metrics

Lingxiao Huang, Jian Li

IEEE COMPUTER SOC2018