On a registration-based approach to sensor network localization

We consider a registration-based approach for localizing sensor networks from range measurements. This is based on the assumption that one can find overlapping cliques spanning the network. That is, for each sensor, one can identify geometric neighbors for which all inter-sensor ranges are known. Such cliques can be efficiently localized using multidimensional scaling. However, since each clique is localized in some local coordinate system, we are required to register them in a global coordinate system. In other words, our approach is based on transforming the localization problem into a problem of registration. In this context, the main contributions are as follows. First, we describe an efficient method for partitioning the network into overlapping cliques. Second, we study the problem of registering the localized cliques, and formulate a necessary rigidity condition for uniquely recovering the global sensor coordinates. In particular, we present a method for efficiently testing rigidity, and a proposal for augmenting the partitioned network to enforce rigidity. A recently proposed semidefinite relaxation of global registration is used for registering the cliques. We present simulation results on random and structured sensor networks to demonstrate that the proposed method compares favourably with state-of-the-art methods in terms of run-time, accuracy, and scalability

On the Composition of Randomized Query Complexity and Approximate Degree

For any Boolean functions $f$ and $g$, the question whether $R(f\circ g) = \tilde{\Theta}(R(f)R(g))$, is known as the composition question for the randomized query complexity. Similarly, the composition question for the approximate degree asks whether $\widetilde{deg}(f\circ g) = \tilde{\Theta}(\widetilde{deg}(f)\cdot\widetilde{deg}(g))$. These questions are two of the most important and well-studied problems, and yet we are far from answering them satisfactorily. It is known that the measures compose if one assumes various properties of the outer function $f$ (or inner function $g$). This paper extends the class of outer functions for which $\text{R}$ and $\widetilde{\text{deg}}$ compose. A recent landmark result (Ben-David and Blais, 2020) showed that $R(f \circ g) = \Omega(noisyR(f)\cdot R(g))$. This implies that composition holds whenever noisyR(f) = \Tilde{\Theta}(R(f)). We show two results: (1)When $R(f) = \Theta(n)$, then $noisyR(f) = \Theta(R(f))$. (2) If $\text{R}$ composes with respect to an outer function, then $\text{noisyR}$ also composes with respect to the same outer function. On the other hand, no result of the type $\widetilde{deg}(f \circ g) = \Omega(M(f) \cdot \widetilde{deg}(g))$ (for some non-trivial complexity measure $M(\cdot)$) was known to the best of our knowledge. We prove that $\widetilde{deg}(f\circ g) = \widetilde{\Omega}(\sqrt{bs(f)} \cdot \widetilde{deg}(g)),$ where $bs(f)$ is the block sensitivity of $f$. This implies that $\widetilde{\text{deg}}$ composes when $\widetilde{\text{deg}}(f)$ is asymptotically equal to $\sqrt{\text{bs}(f)}$. It is already known that both $\text{R}$ and $\widetilde{\text{deg}}$ compose when the outer function is symmetric. We also extend these results to weaker notions of symmetry with respect to the outer function

On the Composition of Randomized Query Complexity and Approximate Degree

For any Boolean functions f and g, the question whether R(f?g) = ??(R(f) ? R(g)), is known as the composition question for the randomized query complexity. Similarly, the composition question for the approximate degree asks whether deg?(f?g) = ??(deg?(f)?deg?(g)). These questions are two of the most important and well-studied problems in the field of analysis of Boolean functions, and yet we are far from answering them satisfactorily. It is known that the measures compose if one assumes various properties of the outer function f (or inner function g). This paper extends the class of outer functions for which R and deg? compose. A recent landmark result (Ben-David and Blais, 2020) showed that R(f?g) = ?(noisyR(f)? R(g)). This implies that composition holds whenever noisyR(f) = ??(R(f)). We show two results: 1. When R(f) = ?(n), then noisyR(f) = ?(R(f)). In other words, composition holds whenever the randomized query complexity of the outer function is full. 2. If R composes with respect to an outer function, then noisyR also composes with respect to the same outer function. On the other hand, no result of the type deg?(f?g) = ?(M(f) ? deg?(g)) (for some non-trivial complexity measure M(?)) was known to the best of our knowledge. We prove that deg?(f?g) = ??(?{bs(f)} ? deg?(g)), where bs(f) is the block sensitivity of f. This implies that deg? composes when deg?(f) is asymptotically equal to ?{bs(f)}. It is already known that both R and deg? compose when the outer function is symmetric. We also extend these results to weaker notions of symmetry with respect to the outer function

Tight Chang’s-lemma-type bounds for Boolean functions

Chang’s lemma (Duke Mathematical Journal, 2002) is a classical result in mathematics, with applications spanning across additive combinatorics, combinatorial number theory, analysis of Boolean functions, communication complexity and algorithm design. For a Boolean function f that takes values in {-1, 1} let r(f) denote its Fourier rank (i.e., the dimension of the span of its Fourier support). For each positive threshold t, Chang’s lemma provides a lower bound on δ(f):= Pr[f(x) = -1] in terms of the dimension of the span of its characters with Fourier coefficients of magnitude at least 1/t. In this work we examine the tightness of Chang’s lemma with respect to the following three natural settings of the threshold: the Fourier sparsity of f, denoted k(f), the Fourier max-supp-entropy of f, denoted k′(f), defined to be the maximum value of the reciprocal of the absolute value of a non-zero Fourier coefficient, the Fourier max-rank-entropy of f, denoted k′′(f), defined to be the minimum t such that characters whose coefficients are at least 1/t in magnitude span a r(f)-dimensional space. In this work we prove new lower bounds on δ(f) in terms of the above measures. One of our lower bounds, δ(f) = Ω (r(f)2/(k(f) log2 k(f))), subsumes and refines the previously best known upper bound r(f) = O(pk(f) log k(f)) on r(f) in terms of k(f) by Sanyal (Theory of Computing, 2019). We improve upon this bound and show r(f) = O(pk(f)δ(f) log k(f)). Another lower bound, δ(f) = Ω (r(f)/(k′′(f) log k(f))), is based on our improvement of a bound by Chattopadhyay, Hatami, Lovett and Tal (ITCS, 2019) on the sum of absolute values of level-1 Fourier coefficients in terms of F2-degree. We further show that Chang’s lemma for the above-mentioned choices of the threshold is asymptotically outperformed by our bounds for most settings of the parameters involved. Next, we show that our bounds are tight for a wide range of the parameters involved, by constructing functions witnessing their tightness. All the functions we construct are modifications of the Addressing function, where we replace certain input variables by suitable functions. Our final contribution is to construct Boolean functions f for which our lower bounds asymptotically match δ(f), and for any choice of the threshold t, the lower bound obtained from Chang’s lemma is asymptotically smaller than δ(f). Our results imply more refined deterministic one-way communication complexity upper bounds for XOR functions. Given the wide-ranging application of Chang’s lemma to areas like additive combinatorics, learning theory and communication complexity, we strongly feel that our refinements of Chang’s lemma will find many more applications

An Inscribed Bronze Sculpture of a Buddha in bhadrāsana at Museum Ranggawarsita in Semarang (Central Java, Indonesia)

A fine bronze Buddha image (ca. 9th century), from Rejoso near Candi Plaosan in Central Java province, was recently found in storage at the Museum Ranggawarsita in Semarang. The Buddha is seated in bhadrāsana, the posture with two legs pendant, and with the hands in dharmacakramudrā, an iconographic type frequently found in South Asia and Java during the second half of the first millennium. This bronze is most remarkable for the Sanskrit inscription written on its back in a northeastern Indian script. The inscription cites the common ye dharmāḥ formula in unique combination with the heart mantra jinajik. Our study deals first with the artistic style and iconography of the bronze image. We then discuss the inscription and its paleographic features, and review the textual sources in which this special mantra occurs. Overall, we are concerned with questions of provenance, dating, and the controversial issue of the identification of the Buddha who is represented. We cautiously propose to consider the inscribed bronze sculpture as embedding a universal and imperial form of Buddhahood, and highlight its significance for our understanding of early tantric Buddhist iconography and concepts in ancient Java and beyond.Une délicate sculpture de Buddha en bronze (ca. IXe siècle), provenant de Rejoso, près du Candi Plaosan dans la province de Java Centre, fut récemment trouvée dans les réserves du musée Ranggawarsita à Semarang. Le Buddha est assis en bhadrāsana, la posture avec les deux jambes pendantes, et les deux mains en dharmacakramudrā, soit un type iconographique fréquent en Asie du Sud et à Java au cours de la deuxième moitié du premier millénaire. Ce bronze est surtout remarquable par le texte en sanskrit inscrit au dos, écrit dans un alphabet du Nord-Est de l'Inde. L'inscription cite la formule ye dharmāḥ, à laquelle s’ajoute de manière exceptionnelle le mantra essentiel (hr̥dayamantra) jinajik. Notre étude s’attache d’abord à analyser le style et l’iconographie de ce Buddha en bronze. Nous examinons ensuite l’inscription et ses caractéristiques paléographiques. Nous étudions enfin les sources textuelles dans lesquelles ce mantra spécifique apparaît. Notre examen de l’œuvre et de son inscription tente de résoudre les questions de provenance et de datation, ainsi que celle, controversée, de l’identité du Buddha représenté. Nous proposons de voir dans cette sculpture en bronze une manifestation universelle et impériale de la bouddhéité. Selon nous, cette pièce inscrite permet de mieux comprendre certains concepts de l’iconographie tantrique bouddhique à date ancienne, à Java et ailleurs.Griffiths Arlo, Revire Nicolas, Sanyal Rajat. An Inscribed Bronze Sculpture of a Buddha in bhadrāsana at Museum Ranggawarsita in Semarang (Central Java, Indonesia). In: Arts asiatiques, tome 68, 2013. pp. 3-26

Avalokiteśvara of the “Three and a Half Syllables”: A Note on the Heart-Mantra Ārolik in India

本 論 は 、 紀 元 後 第 一 千 年 紀 の 後 半 に 遡 る 三 つ の 菩 薩 像 に 刻 ま れ た 珍 し い 碑 文 を 検 討 す る も の で あ る 。 一 つ は ウ ッ タ ラ ・ プ ラ デ ー シ ュ 州 の Sarnath で 、 後 の 二 つ は ビ ハ ー ル 州 の Telhara と Bargaon で 出 土 し た 。 こ れ ら の 彫 像 に は 、 観 世 音 菩 薩 の 「 三 字 半 心 真 言 」 、 す な わ ち 「 ア ー ロ ー リ ク 」 ārolik が 刻 ま れ て い る 。 こ の 心 真 言 は 蓮 華 部 に 属 し て い る 。 ビ ハ ー ル 出 土 の 断 片 は 、 お そ ら く 六 臂 の 不 空 羂 索 観 音 を 表 わ す も の で あ る 。 一 方 、 Bargaon の 像 に 刻 ま れ た 文 面 は 、 南 ア ジ ア で 作 ら れ 、 第 一 千 年 紀 半 ば か ら 後 半 に か け て 東 ア ジ ア に 伝 え ら れ た 不 空 羂 索 心 呪 を 記 す 唯 一 の サ ン ス ク リ ッ ト 碑 文 で あ る 。 Ārolik と い う 心 真 言 は ま た 、 サ ン ス ク リ ッ ト 、 漢 文 、 チ ベ ッ ト 語 で 残 さ れ た 密 教 文 献 で も 見 る こ と が で き る 。 本 論 の 終 わ り で は 、 古 い イ ン ド の 仏 教 美 術 で 、 観 音 像 を 同 定 す る こ と の よ り 広 い 意 義 に つ い て 考 え る こ と に す る 。This article examines rare epigraphical evidence engraved on three inscribed Bodhisattva sculptures dated to the second half of the first millennium from Sarnath, in Uttar Pradesh, Telhara, and Bargaon, in Bihar. The inscriptions contain the heart-mantra ārolik, i.e., the “three and a half syllables” connected to Avalokiteśvara and the “Lotus Family.” The fragments from Bihar probably depict a six-armed Amoghapāśa, a specific iconographic form of Avalokiteśvara, while the Bargaon inscription is the only identified occurrence in Sanskrit epigraphy of the Amoghapāśahr̥dayadhāraṇī, composed in South Asia and transmitted to East Asia in the mid-to-late first millennium. The heart-mantra ārolik is also known in esoteric and tantric Buddhist sources still preserved in Sanskrit originals or Chinese and Tibetan translations. Our study concludes on the broader implications for the identification of Avalokiteśvara in early Indian Buddhist art.Cet article examine de rares inscriptions gravées sur trois sculptures de bodhisattva datées de la seconde moitié du premier millénaire et originaires de Sarnath en Uttar Pradesh, de Telhara et de Bargaon au Bihar. Les inscriptions contiennent le mantra du cœur d’Avalokiteśvara, soit ārolik, les « trois syllabes et demie » liées à la « famille du lotus ». Les fragments du Bihar représentent probablement un Amoghapāśa à six bras, forme iconographique spécifique d’Avalokiteśvara, alors que l’inscription de Bargaon est la seule occurrence identifiée en épigraphie sanskrite de l’Amoghapāśahr̥dayadhāraṇī, composée en Asie du Sud et transmise en Extrême-Orient vers la fin du premier millénaire. Le mantra du cœur ārolik est aussi connu dans les sources ésotériques et tantriques bouddhiques encore préservées en sanskrit ou dans des traductions chinoises et tibétaines. Notre étude conclut sur les répercussions plus vastes concernant l’identification d’Avalokiteśvara dans le premier art bouddhique indien.本 文 考 察 了 三 座 菩 薩 雕 像 上 的 罕 見 銘 文 ， 這 些 雕 像 製 作 於 第 一 個 千 年 的 後 半 期 ， 分 別 出 自 北 方 邦 的 鹿 野 苑 、 比 哈 爾 邦 的 泰 爾 哈 拉 和 伯 爾 岡 。 銘 文 內 容 涉 及 阿 嚕 力 迦 心 真 言 ， 此 乃 觀 自 在 菩 薩 蓮 花 部 三 字 半 真 言 。 比 哈 爾 邦 雕 像 殘 件 的 原 型 極 有 可 能 是 六 臂 不 空 羂 索 觀 音 ， 即 觀 自 在 菩 薩 的 化 身 之 一 。 此 外 ， 伯 爾 岡 雕 像 銘 文 是 唯 一 確 定 的 梵 語 銘 刻 的 不 空 羂 索 心 陀 羅 尼 ， 此 陀 羅 尼 創 作 於 南 亞 ， 並 於 第 一 個 千 年 中 後 期 傳 至 東 亞 。 阿 嚕 力 迦 心 真 言 亦 可 見 於 密 教 文 獻 中 ， 梵 語 原 本 和 漢 語 、 藏 語 譯 本 中 均 有 記 載 。 本 文 研 究 結 果 表 明 ， 早 期 印 度 佛 教 美 術 中 觀 自 在 菩 薩 的 鑒 別 取 得 了 進 一 步 的 進 展 。Revire Nicolas, Sanyal Rajat, Giebel Rolf. Avalokiteśvara of the “Three and a Half Syllables”: A Note on the Heart-Mantra Ārolik in India. In: Arts asiatiques, tome 76, 2021. pp. 5-30

CONSENSUS OPTIMIZATION FOR DISTRIBUTED REGISTRATION

We consider the problem of jointly registering multiple point sets using rigid transforms. We propose a distributed algorithm based on consensus optimization for the least-squares formulation of this problem. In each iteration, the computation is distributed among the point sets and the results are averaged. For each point set, the dominant cost per iteration is the SVD of a square matrix of size d, where d is the ambient dimension. Existing methods for joint registration are either centralized or perform the optimization sequentially. The proposed algorithm is naturally more scalable than these methods. As an application, we integrate the proposed algorithm within a divide-and-conquer approach for sensor network localization. In particular, we are able to localize very large networks, which are beyond the scope of most existing localization methods

A CASE OF PRIMARY UTERINE LYMPHOMA

A post menopausal lady presented with lump lower abdomen and bleeding per vaginum. USG revealed diffuse enlargement of the uterus. On hysterectomy, a grossly enlarged uterus with cystic left ovary were found. Hysterectomy was done and uterus with bilateral adnexa submitted for histopathological examination. Microscopic examination of the body of uterus revealed sheets of small lymphoid cells were found to replace the endo- and myo-metrium. These cells have small nuclei with clumped chromatin, and no prominence of nucleoli. They are not forming lymphoid follicles or germinal centers. Similar lymphoid cells were also found in the left ovary admixed with ovarian stroma. On IHC these cells were found to be CD45, CD20, CD23 positive, and negative for CD3 , CK and SMA. The case is diagnosed as a primary small lymphocytic lymphoma of uterus with left ovarian spread

A Scalable ADMM Algorithm for Rigid Registration

A fundamental problem that comes up in computer vision, image processing, manifold learning, and sensor networks is that of registering multiple point sets using rigid transforms. A standard result in this regard is that the least-square formulation of the registration problem admits a closed-form solution for two point sets. However, since the group of rigid transforms is not convex, solving the least-square optimization for multiple point sets is computationally challenging. It was recently demonstrated that the least-square formulation can be relaxed into a tractable semidefinite program, and that the relaxation is provably tight under certain assumptions. The difficulty is that standard solvers for semidefinite programming (e.g., interior-point solvers) cannot be scaled to handle large-sized problems. In this letter, we propose an iterative solver based on variable splitting and the alternating direction method of multipliers. Since each iteration essentially involves an eigendecomposition, the proposed solver can be scaled to problems that are beyond the reach of interior-point solvers. We present results on simulated and real data to demonstrate the potential of the solver