Convex Relaxation for Low-Dimensional Representation: Phase Transitions and Limitations

There is a growing interest in taking advantage of possible patterns and structures in data so as to extract the desired information and overcome the curse of dimensionality. In a wide range of applications, including computer vision, machine learning, medical imaging, and social networks, the signal that gives rise to the observations can be modeled to be approximately sparse and exploiting this fact can be very beneficial. This has led to an immense interest in the problem of efficiently reconstructing a sparse signal from limited linear observations. More recently, low-rank approximation techniques have become prominent tools to approach problems arising in machine learning, system identification and quantum tomography.

In sparse and low-rank estimation problems, the challenge is the inherent intractability of the objective function, and one needs efficient methods to capture the low-dimensionality of these models. Convex optimization is often a promising tool to attack such problems. An intractable problem with a combinatorial objective can often be "relaxed" to obtain a tractable but almost as powerful convex optimization problem. This dissertation studies convex optimization techniques that can take advantage of low-dimensional representations of the underlying high-dimensional data. We provide provable guarantees that ensure that the proposed algorithms will succeed under reasonable conditions, and answer questions of the following flavor:

More specifically, i) Focusing on linear inverse problems, we generalize the classical error bounds known for the least-squares technique to the lasso formulation, which incorporates the signal model. ii) We show that intuitive convex approaches do not perform as well as expected when it comes to signals that have multiple low-dimensional structures simultaneously. iii) Finally, we propose convex relaxations for the graph clustering problem and give sharp performance guarantees for a family of graphs arising from the so-called stochastic block model. We pay particular attention to the following aspects. For i) and ii), we aim to provide a general geometric framework, in which the results on sparse and low-rank estimation can be obtained as special cases. For i) and iii), we investigate the precise performance characterization, which yields the right constants in our bounds and the true dependence between the problem parameters.

Classical correlated imaging from the perspective of coherent-mode representation

We use coherent-mode representation of partially coherent fields to analyze correlated imaging with classical light sources. This formalism is very useful to study the imaging quality. By decomposing the unknown object as the superposition of different coherent modes, the components corresponding to small eigenvalues cannot be well imaged. The generated images depend crucially on the distribution of the eigenvalues of the coherent-mode representation of the source and the decomposition coefficients of the objects. Three kinds of correlated imaging schemes are analyzed numerically.

Análise comparativa in vivo da detecção de lesões de cárie oclusal e proximal por meio de exame visual (ICDAS), exame radiográfico e transiluminação por fibra ótica (FOTI)

O objetivo do presente trabalho foi avaliar in vivo a detecção de cárie através do exame visual ICDAS, transiluminação por fibra ótica combinado ao ICDAS e exame radiográfico. Um total de 2.279 superfícies proximais e cicatrículas e fissuras em incisivos superiores, pré-molares e molares permanentes e 272 superfícies em molares decíduos em72 escolares (8 a 18 anos) foram avaliadas por um examinador treinado. Os sete escores para detecção de cárie primária do sistema visual ICDAS foram aplicados. Dois equipamentos de transiluminação por fibra ótica foram avaliados: FOTI Schott (SCH), com ponta de fibra ótica com 0,5mm de diâmetro, e FOTI Microlux (MIC), com diâmetro da ponta 3 mm. Durante o exame combinado FOTI/ICDAS, a fibra ótica era utilizada tanto para iluminar quanto para transiluminar a superfície sob avaliação. O exame radiográfico (RX) consistiu de radiografias interproximais posteriores e periapicais anteriores. Os exames foram realizados em consultório odontológico após escovação supervisionada. No primeiro dia de exame, o exame visual utilizando o ICDAS era realizado e em seguida, o exame combinado ao MIC ou SCH. Logo após era realizado o exame radiográfico. Após uma semana, novamente o ICDAS era realizado, e em seguida o exame combinado com o equipamento de FOTI não utilizado na semana anterior. Os exames foram repetidos em 10 pacientes após intervalo mínimo de uma semana para avaliação da reprodutibilidade intra-examinador, a qual apresentou valores de 0,95 (ICDAS), 0,94 (MIC), 0,95 (SCH) e 0,99 (RX) pelo kappa ponderado. Em cicatrículas e fissuras de permanentes, o RX julgou que um número maior de superfícies apresentava lesão em dentina (53) do que os outros métodos (34 a 36); porém não detectou nenhuma lesão em esmalte, as quais foram identificadas pelo ICDAS (94), SCH (107) e MIC (91). Em proximais permanentes, a transiluminação por fibra ótica identificou maior número de proximais como lesão em esmalte - 150 (SCH) e 139 (MIC) - do que o exame visual (106), enquanto o RX identificou somente 43. Em oclusais de decíduos, os quatro métodos julgaram um número aproximadamente similar de superfícies sem lesão (52 a 59) ou com lesão em dentina (21 a 26), assim como para lesões proximais em dentina (31 a 36). Entretanto um número reduzido de lesões proximais decíduas em esmalte foi julgado pelo exame radiográfico (3) em comparação com os outros métodos (15 a 16). Em decíduos, o ICDAS e o FOTI combinado ao exame visual julgaram maior número de lesões proximais em esmalte que o exame radiográfico, sendo que número similar de lesões em dentina foram classificadas pelos quatro métodos em oclusais e proximais de molares decíduos. Em cicatrículas e fissuras de permanentes, tanto o exame visual ICDAS quanto sua combinação aos dois equipamentos de transiluminação apresentaram maior similaridade de superfícies julgadas como lesão em esmalte ou como lesão em dentina, enquanto o exame radiográfico classificou mais superfícies como lesão em dentina e nenhuma como lesão em esmalte. A adição da transiluminação por fibra ótica ao exame visual aumentou em um terço a detecção das lesões cariosas proximais julgadas em dentina pelo ICDAS isoladamente e aproximadamente quadruplicou o número daquelas assim classificadas pela avaliação radiográfica em permanentes.

Visual pattern recognition network: Its training algorithm and its optoelectronic architecture

A visual pattern recognition network and its training algorithm are proposed. The network constructed of a one-layer morphology network and a two-layer modified Hamming net. This visual network can implement invariant pattern recognition with respect to image translation and size projection. After supervised learning takes place, the visual network extracts image features and classifies patterns much the same as living beings do. Moreover we set up its optoelectronic architecture for real-time pattern recognition. (C) 1996 Optical Society of America

Hit-or-miss representation of a pattern spectrum and its optical implementation

We demonstrate that a pattern spectrum can be decomposed into the union of hit-or-miss transforms with respect to a series of structure-element pairs. Moreover we use a Boolean-logic function to express the pattern spectrum and show that the Boolean-logic representation of a pattern spectrum is composed of hit-or-miss min terms. The optical implementation of a pattern spectrum is based on an incoherent optical correlator with a feedback operation. (C) 1996 Optical Society of America

Studies of the representation of interatomic distances in molecules and crystal as sum of atomic radii

The electron diffraction investigation of the following compounds has been carried out: sulfur, sulfur nitride, realgar, arsenic trisulfide, spiropentane, dimethyltrisulfide, cis and trans lewisite, methylal, and ethylene glycol.

The crystal structures of the following salts have been determined by x-ray diffraction: silver molybdateand hydrazinium dichloride.

Suggested revisions of the covalent radii for B, Si, P, Ge, As, Sn, Sb, and Pb have been made, and values for the covalent radii of Al, Ga, In, Ti, and Bi have been proposed.

The Schomaker-Stevenson revision of the additivity rule for single covalent bond distances has been used in conjunction with the revised radii. Agreement with experiment is in general better with the revised radii than with the former radii and additivity.

The principle of ionic bond character in addition to that present in a normal covalent bond has been applied to the observed structures of numerous molecules. It leads to a method of interpretation which is at least as consistent as the theory of multiple bond formation.

The revision of the additivity rule has been extended to double bonds. An encouraging beginning along these lines has been made, but additional experimental data are needed for clarification.

Code conversion from signed-digit to complement representation based on look-ahead optical logic operations

We present, for the first time to our knowledge, a generalized lookahead logic algorithm for number conversion from signed-digit to complement representation. By properly encoding the signed-digits, all the operations are performed by binary logic, and unified logical expressions can be obtained for conversion from modified-signed-digit (MSD) to 2's complement, trinary signed-digit (TSD) to 3's complement, and quarternary signed-digit (QSD) to 4's complement. For optical implementation, a parallel logical array module using an electron-trapping device is employed and experimental results are shown. This optical module is suitable for implementing complex logic functions in the form of the sum of the product. The algorithm and architecture are compatible with a general-purpose optoelectronic computing system. (C) 2001 Society of Photo-Optical Instrumentation Engineers.