La k-PDTM : un coreset pour l'inférence géométrique

Analyzing the sub-level sets of the distance to a compact sub-manifold of R d is a common method in TDA to understand its topology. The distance to measure (DTM) was introduced by Chazal, Cohen-Steiner and Mérigot in [7] to face the non-robustness of the distance to a compact set to noise and outliers. This function makes possible the inference of the topology of a compact subset of R d from a noisy cloud of n points lying nearby in the Wasserstein sense. In practice, these sub-level sets may be computed using approximations of the DTM such as the q-witnessed distance [10] or other power distance [6]. These approaches lead eventually to compute the homology of unions of n growing balls, that might become intractable whenever n is large. To simultaneously face the two problems of large number of points and noise, we introduce the k-power distance to measure (k-PDTM). This new approximation of the distance to measure may be thought of as a k-coreset based approximation of the DTM. Its sublevel sets consist in union of k-balls, k << n, and this distance is also proved robust to noise. We assess the quality of this approximation for k possibly dramatically smaller than n, for instance k = n 1 3 is proved to be optimal for 2-dimensional shapes. We also provide an algorithm to compute this k-PDTM.L'analyse des sous niveaux de la fonction distance à une variété compacte de R d est très fréquente en analyse topologique des données, avec pour objectif d'en comprendre la topologie. La distance à la mesure (DTM) a été introduite par Chazal, Cohen-Steiner et Mérigot avec l'objectif de remédier au caractère non robuste au bruit et aux données aberrantes de la distance à un compact. Cette fonction rend possible l'inférence de la topologie d'un sous-ensemble compact de R d à partir d'un nuage de n points tirés dans un voisinage proche de la sous-variété au sens de Wasserstein. En pratique, les sous-ensembles de niveau de cette fonction peuvent être estimés en utilisant des approximations de la DTM tels que la q-witnessed distance ou d'autres fonctions puissance. Ces approches reviennent à calculer l'homologie de l'union de n boules, ce qui devient impossible en pratique lorsque n devient trop grand. Afin de traiter le problème du grand nombre de points et du bruit, on introduit la fonction k-puissance distance à la mesure (k-PDTM). Cette nouvelle approximation de la distance à la mesure peut être vue une approximation de la DTM s'appuyant sur un $k$-coreset. Ses sous-niveaux seront alors des unions de k boules pour k<<n, et cette fonction est également robuste au bruit. On étudie la qualité de cette approximation lorsque k est très petit par rapport à n. Par exemple, le choix de k=n^{1/3} est optimal pour des formes en dimension 2. On fournit également un algorithme pour calculer cette fonction k-PDTM

Against the Nihilism of Suffering and Death: Richard E. K. Kim and His Works

This article examines the life and works of Richard E. K. Kim (1932–2009), a first-generation Korean diasporic writer in the United States. It focuses on how Kim struggled to overcome the nihilism of suffering and death that derived from colonialism and the Korean War through his literary works. Kim witnessed firsthand these two major historical events, which caused irrevocable psychological and physical damage to many people of his generation. In his autobiographical fiction, he conveys painful memories of the events by reviving the voices of people in that era. What his works offer us goes beyond vivid memories of the past, however; they also present the power of forgiveness as a condition to overcome the nihilism of suffering and death. Remembrance and forgiveness are, therefore, two major thematic pillars of his works that enable us to connect to these difficult and traumatic times. These themes are portrayed in such a gripping way mainly because Kim tried to maintain a certain distance—an emotional and linguistic distance—from the familiar, in order to elucidate the reality of the human condition: an ontological position of the exile from which he produced his works. This article argues that Kim’s works provide us the possibility to transcend the nihilism of historical trauma through articulating the meaning of remembrance and forgiveness from his self-assumed position of exile. Keywords: Richard E. K. Kim, diasporic writer, Lost Names, The Martyred, Searching for Lost Times, Japanese colonialism, Korean War, remembrance, forgivenes

Deterministic Digital Clustering of Wireless Ad Hoc Networks

We consider deterministic distributed communication in wireless ad hoc networks of identical weak devices under the SINR model without predefined infrastructure. Most algorithmic results in this model rely on various additional features or capabilities, e.g., randomization, access to geographic coordinates, power control, carrier sensing with various precision of measurements, and/or interference cancellation. We study a pure scenario, when no such properties are available. As a general tool, we develop a deterministic distributed clustering algorithm. Our solution relies on a new type of combinatorial structures (selectors), which might be of independent interest. Using the clustering, we develop a deterministic distributed local broadcast algorithm accomplishing this task in $O(\Delta \log^*N \log N)$ rounds, where $\Delta$ is the density of the network. To the best of our knowledge, this is the first solution in pure scenario which is only polylog$(n)$ away from the universal lower bound $\Omega(\Delta)$, valid also for scenarios with randomization and other features. Therefore, none of these features substantially helps in performing the local broadcast task. Using clustering, we also build a deterministic global broadcast algorithm that terminates within $O(D(\Delta + \log^* N) \log N)$ rounds, where $D$ is the diameter of the network. This result is complemented by a lower bound $\Omega(D \Delta^{1-1/\alpha})$, where $\alpha > 2$ is the path-loss parameter of the environment. This lower bound shows that randomization or knowledge of own location substantially help (by a factor polynomial in $\Delta$) in the global broadcast. Therefore, unlike in the case of local broadcast, some additional model features may help in global broadcast

Experimental Demonstration of Macroscopic Quantum Coherence in Gaussian States

We witness experimentally the presence of macroscopic coherence in Gaussian quantum states using a recently proposed criterion (E.G. Cavalcanti and M. Reid, Phys. Rev. Lett. 97, 170405 (2006)). The macroscopic coherence stems from interference between macroscopically distinct states in phase space and we prove experimentally that even the vacuum state contains these features with a distance in phase space of $0.51\pm0.02$ shot noise units (SNU). For squeezed states we found macroscopic superpositions with a distance of up to $0.83\pm0.02$ SNU. The proof of macroscopic quantum coherence was investigated with respect to squeezing and purity of the states.Comment: 5 pages, 6 figure