Low rank tensor recovery via iterative hard thresholding

We study extensions of compressive sensing and low rank matrix recovery (matrix completion) to the recovery of low rank tensors of higher order from a small number of linear measurements. While the theoretical understanding of low rank matrix recovery is already well-developed, only few contributions on the low rank tensor recovery problem are available so far. In this paper, we introduce versions of the iterative hard thresholding algorithm for several tensor decompositions, namely the higher order singular value decomposition (HOSVD), the tensor train format (TT), and the general hierarchical Tucker decomposition (HT). We provide a partial convergence result for these algorithms which is based on a variant of the restricted isometry property of the measurement operator adapted to the tensor decomposition at hand that induces a corresponding notion of tensor rank. We show that subgaussian measurement ensembles satisfy the tensor restricted isometry property with high probability under a certain almost optimal bound on the number of measurements which depends on the corresponding tensor format. These bounds are extended to partial Fourier maps combined with random sign flips of the tensor entries. Finally, we illustrate the performance of iterative hard thresholding methods for tensor recovery via numerical experiments where we consider recovery from Gaussian random measurements, tensor completion (recovery of missing entries), and Fourier measurements for third order tensors.Comment: 34 page

Low-rank Tensor Recovery

Low-rank tensor recovery is an interesting subject from both the theoretical and application point of view. On one side, it is a natural extension of the sparse vector and low-rank matrix recovery problem. On the other side, estimating a low-rank tensor has applications in many different areas such as machine learning, video compression, and seismic data interpolation. In this thesis, two approaches are introduced. The first approach is a convex optimization approach and could be considered as a tractable extension of $ell_1$-minimization for sparse vector and nuclear norm minimization for matrix recovery to tensor scenario. It is based on theta bodies – a recently introduced tool from real algebraic geometry. In particular, theta bodies of appropriately defined polynomial ideal correspond to the unit-theta norm balls. These unit-theta norm balls are relaxations of the unit-tensor-nuclear norm ball. Thus, in this case, we consider a canonical tensor format. The method requires computing the reduced Groebner basis (with respect to the graded reverse lexicographic ordering) of the appropriately defined polynomial ideal. Numerical results for third-order tensor recovery via $theta_1$-norm are provided. The second approach is a generalization of iterative hard thresholding algorithm for sparse vector and low-rank matrix recovery to tensor scenario (tensor IHT or TIHT algorithm). Here, we consider the Tucker format, the tensor train decomposition, and the hierarchical Tucker decomposition. The analysis of the algorithm is based on a version of the restricted isometry property (tensor RIP or TRIP) adapted to the tensor decomposition at hand. We show that subgaussian measurement ensembles satisfy TRIP with high probability under an almost optimal condition on the number of measurements. Additionally, we show that partial Fourier maps combined with random sign flips of the tensor entries satisfy TRIP with high probability. Under the assumption that the linear operator satisfies TRIP and under an additional assumption on the thresholding operator, we provide a linear convergence result for the TIHT algorithm. Finally, we present numerical results on low-Tucker-rank third-order tensors via partial Fourier maps combined with random sign flips of tensor entries, tensor completion, and Gaussian measurement ensembles

The Bromination of 2-Furanilide

In connection with studies of the reactions of some thioamides, to be published later, we were led to investigate the bromination of 2-furanilide, hitherto not mentioned in the literature

PREMISE OF ENCOURAGING AUTONOMY IN CHILDREN\u27S WORK AND OPINION

Cilj ovoga rada je uočavanje određenih metoda koje bi dovele do autonomije djece u njihovim aktivnostima, a samim time i do razvoja kritičkog mišljenja. Pripremajući poticajno prostorno-materijalno okruženje, nastoji se imati na umu individualne sposobnosti djece i individualne interese prema pojedinim područjima. Podjelom prostora na mnoštvo različitih centara djeca samostalno biraju individualni pristup ili rad u manjim grupama. Podržavanjem samoiniciranih aktivnosti djece proizašlih iz poticajnog okruženja djeci se omogućava stjecanje znanja, konstruiranje znanja, samostalno donošenje odluka, smjer razvoja aktivnosti te preuzimanje odgovornosti za realizaciju iste. Postavljajući pitanja, nastoji se djetetu ukazati na zanimanje za ono što radi naglašavanjem važnosti djetetove aktivnosti i davanje podrške za daljnji nivo djetetova razmišljanja. Na taj se način potiče djetetovo samopouzdanje i samostalnost. Osim osiguravanja prostorno-materijalnog okruženja i podržavanjem samoiniciranih dječjih aktivnosti, odgojitelj podržava i prepoznaje rad djece na projektima proizašlih iz njihovih interesa. Rad djece na projektu, kao metoda apsolutne autonomije, omogućava atmosferu koja će rezultirati opuštenom realizacijom projekta, uvažavajući dječje mišljenje o daljnjem razvoju projekta. Rezultati praćenja kroz određeni vremenski period pokazali su da se ovakvim pristupom radu djeca osjećaju prihvaćena, jačaju samopouzdanje, konstruiraju vlastito razumijevanje, uče se dijalogu, neometano iznose mišljenja i stavove te postavljaju nove hipoteze i sve više ulaze u druge nivoe nadogradnje cjelokupnog razvoja

Anemia and inflammatory bowel diseases

Cilj istraživanja: Cilj ovog istraživanja bio je utvrditi učestalost i vrstu anemije u odraslih osoba oboljelih od upalnih bolesti crijeva s područja istočne Hrvatske te utvrditi postoji li povezanost između pojavnosti anemije te vrijednosti C-reaktivnog proteina u krvi oboljelih kao neizravnog pokazatelja aktivnosti upale. Ispitanici i metode: Provedeno istraživanje ustrojeno je po načelu cross-sectional studije odnosno kao tipična metoda istraživanja presjeka ili prevalencije. U istraživanje je uključeno ukupno 78 oboljelih od upalnih bolesti crijeva s područja istočne Hrvatske u rasponu od 22 do 69 godina. Korištene su sljedeće laboratorijske analize za dokazivanje anemije: određivanje željeza u serumu, transferina, feritina, TIBC-a,UIBC-a, C-reaktivnog proteina, kompletne krvne slike i retikulocita. Rezultati: Ovim istraživanjem utvrđena je prisutnost anemije kod 24,4 % ispitanika. Prema vrsti anemije, u 63,2 % slučajeva riječ je o sideropeničnoj anemiji, a u 36,8 % slučajeva riječ je o anemiji kronične bolesti. Nema statistički značajne razlike u učestalosti i vrsti anemije između oboljelih od Crohnove bolesti te oboljelih od ulceroznog kolitisa. Kod osoba s aktivnom bolešću statistički je značajno češće utvrđeno postojanje anemije u odnosu na osobe s inaktivnom bolešću. Zaključak: Anemija je značajna ekstraintestinalna komplikacija upalnih bolesti crijeva te kako slijedom izostanka liječenja anemije u sklopu upalne bolesti crijeva anemija predstavlja značajan dodatni simptom odnosno komplikaciju kod oboljelih.Objectives: The aim of this study was to determine the frequency and type of anaemia in adult patients with inflammatory bowel disease in the Eastern Croatia. Furthermore, the aim was to examine whether there is correlation between the incidence of anaemia and the value of C-reactive protein in patients’ blood, which could be considered an indirect indicator of the inflammation. Participants and methods: The study is organized as a cross-sectional study, i. e. cross-section and prevalence survey method. The study included 78 patients with IBD from Eastern Croatia 22 – 69 years of age. The following laboratory analyses for the detection of anaemia were used: serum levels of iron, transferrin, ferritin, TIBC, UIBC, C-reactive protein, complete blood count and reticulocyte count. Results: comorbidity present in 24.4% of patients with inflammatory bowel disease in Eastern Croatia. Anaemia was present in 24.4% of the subjects. According to the type of anaemia, in 63.2% of cases it is sideropenic anaemia, and in 36.8% of cases the chronic disease anaemia. Statistically significant differences were not found in the frequency and type of anaemia in patients with Crohn's disease in relation to patients with ulcerative colitis. In patients with active illness, there is a statistically significant increase in the presence of anaemia in relation to patients with inactive disease. Conclusion: Anaemia is a significant extra-intestinal complication of inflammatory bowel disease and in the absence of the treatment of anaemia within the inflammatory bowel disease it is a significant additional symptom or complication in the affected population

A characteristic free approach to secant varieties of triple Segre products

The goal of this short note is to study the secant varieties of the triple Segre product of type (1,a,b) by means of the standard tools of combinatorial commutative algebra. We reprove and extend to arbitrary characteristic results of Landsberg and Weyman regarding the defining ideal and the Cohen-Macaulay property of the secant varieties. Furthermore for these varieties we compute the degree and give a bound for their Castelnuovo-Mumford regularity which is sharp in many cases

The Bromination of 2-Furanilide

In connection with studies of the reactions of some thioamides, to be published later, we were led to investigate the bromination of 2-furanilide, hitherto not mentioned in the literature