A Survey of Geometric Optimization for Deep Learning: From Euclidean Space to Riemannian Manifold

Although Deep Learning (DL) has achieved success in complex Artificial Intelligence (AI) tasks, it suffers from various notorious problems (e.g., feature redundancy, and vanishing or exploding gradients), since updating parameters in Euclidean space cannot fully exploit the geometric structure of the solution space. As a promising alternative solution, Riemannian-based DL uses geometric optimization to update parameters on Riemannian manifolds and can leverage the underlying geometric information. Accordingly, this article presents a comprehensive survey of applying geometric optimization in DL. At first, this article introduces the basic procedure of the geometric optimization, including various geometric optimizers and some concepts of Riemannian manifold. Subsequently, this article investigates the application of geometric optimization in different DL networks in various AI tasks, e.g., convolution neural network, recurrent neural network, transfer learning, and optimal transport. Additionally, typical public toolboxes that implement optimization on manifold are also discussed. Finally, this article makes a performance comparison between different deep geometric optimization methods under image recognition scenarios.Comment: 41 page

Improved integration of information to reduce subsurface model bias

Subsurface modeling deals with data-related issues like cognitive and sampling biases, and model-related challenges including statistical assumptions, misspecification, and algorithmic biases. These challenges introduce four critical implications during subsurface modeling. Firstly, subsurface sampling is subject to sampling bias, which compromises statistical representativeness. Secondly, analog selection methodologies rely on multivariate statistics and expert judgment that overlook spatial information and data dimensionality. Thirdly, subsurface inferential workflows that utilize dimensionality reduction seldom provide repeatable frameworks that maintain model stability and are invariant to Euclidean transformations. Lastly, deep learning methods for dimensionality reduction, characterized as black-box models, lack interpretability and robust evaluation metrics, increasing susceptibility to algorithmic bias. Consequently, neglecting these challenges in subsurface modeling could lead to erroneous predictions, inconsistent inferences, diminished model reliability, and suboptimal decision-making that impacts project economics. This dissertation integrates information within subsurface models to reduce model bias and significantly improve their accuracy, robustness, and generalizability. First, I create spatial declustering methods to debias spatial datasets with single and multiscale preferential sampling in stationary populations. Second, I introduce a novel geostatistics-based machine learning method for identifying subsurface resource analogs that integrate spatial information in subsurface datasets with high dimensionality. Next, I efficiently combine machine learning and computational geometry methods to stabilize lower dimensional spaces for uncertainty quantification and interpretation. Finally, I create a methodology to assess, evaluate, and interpret the stability of deep learning latent feature spaces. These novel methodologies demonstrate the importance of improved techniques for information integration in subsurface modeling and show better results over naïve methods. This results in objective sampling debiasing in spatial stationary populations with single or multiple data scales, improving statistical representativity. Also, the results show better generalization and accurate identification of spatial analogs in high-dimensional datasets. Moreover, the methods yield Euclidean transformation-invariant lower-dimensional spaces, ensuring unique and repeatable solutions that improve model reliability and interpretability, for rational comparisons. Finally, the results indicate that deep learning models for dimensionality reduction exhibit algorithmic biases and instabilities, including sample, structural, and inferential instability, affecting their reliability and interpretability. Together, these innovations ultimately reduce model bias and significantly improve subsurface modeling.Petroleum and Geosystems Engineerin

Unsupervised Machine Learning With Applications to Disease Phenotyping

This dissertation develops two methodologies in unsupervised machine learning (UL), specifically in clustering and data integration, and applies UL methodology to the investigation of phenotypes of knee osteoarthritis (KOA). First, we present novel methodology for assessing the statistical significance of clustering in the important setting of strongly unbalanced cluster sizes, which occur, for example, in the context of rare phenotypes of disease. Second, we develop a new Bayesian data integration model, which brings ideas from the frequentist data integration literature into a Bayesian setting where a posterior distribution provides rich inference and uncertainty quantification. Third, we use biclustering, a UL tool developed for gene expression data, to investigate phenotypes of KOA. Fourth, we present a data integration analysis of the common modes of variation in cartilage in KOA, based on novel cartilage thickness maps and clinical and demographic variables.Doctor of Philosoph

Inferring Human Pose and Motion from Images

As optical gesture recognition technology advances, touchless human computer interfaces of the future will soon become a reality. One particular technology, markerless motion capture, has gained a large amount of attention, with widespread application in diverse disciplines, including medical science, sports analysis, advanced user interfaces, and virtual arts. However, the complexity of human anatomy makes markerless motion capture a non-trivial problem: I) parameterised pose configuration exhibits high dimensionality, and II) there is considerable ambiguity in surjective inverse mapping from observation to pose configuration spaces with a limited number of camera views. These factors together lead to multimodality in high dimensional space, making markerless motion capture an ill-posed problem. This study challenges these difficulties by introducing a new framework. It begins with automatically modelling specific subject template models and calibrating posture at the initial stage. Subsequent tracking is accomplished by embedding naturally-inspired global optimisation into the sequential Bayesian filtering framework. Tracking is enhanced by several robust evaluation improvements. Sparsity of images is managed by compressive evaluation, further accelerating computational efficiency in high dimensional space

Human pose estimation from video and inertial sensors

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Learning from the Artist: Theory and Practice of Example-Based Character Deformation

Movie and game production is very laborious, frequently involving hundreds of person-years for a single project. At present this work is difficult to fully automate, since it involves subjective and artistic judgments. Broadly speaking, in this thesis we explore an approach that works with the artist, accelerating their work without attempting to replace them. More specifically, we describe an “example-based” approach, in which artists provide examples of the desired shapes of the character, and the results gradually improve as more examples are given. Since a character’s skin shape deforms as the pose or expression changes, or particular problem will be termed character deformation. The overall goal of this thesis is to contribute a complete investigation and development of an example-based approach to character deformation. A central observation guiding this research is that character animation can be formulated as a high-dimensional problem, rather than the two- or three-dimensional viewpoint that is commonly adopted in computer graphics. A second observation guiding our inquiry is that statistical learning concepts are relevant. We show that example-based character animation algorithms can be informed, developed, and improved using these observations. This thesis provides definitive surveys of example-based facial and body skin deformation. This thesis analyzes the two leading families of example-based character deformation algorithms from the point of view of statistical regression. In doing so we show that a wide variety of existing tools in machine learning are applicable to our problem. We also identify several techniques that are not suitable due to the nature of the training data, and the high-dimensional nature of this regression problem. We evaluate the design decisions underlying these example-based algorithms, thus providing the groundwork for a ”best practice” choice of specific algorithms. This thesis develops several new algorithms for accelerating example-based facial animation. The first algorithm allows unspecified degrees of freedom to be automatically determined based on the style of previous, completed animations. A second algorithm allows rapid editing and control of the process of transferring motion capture of a human actor to a computer graphics character. The thesis identifies and develops several unpublished relations between the underlying mathematical techniques. Lastly, the thesis provides novel tutorial derivations of several mathematical concepts, using only the linear algebra tools that are likely to be familiar to experts in computer graphics. Portions of the research in this thesis have been published in eight papers, with two appearing in premier forums in the field

Geometric data understanding : deriving case specific features

There exists a tradition using precise geometric modeling, where uncertainties in data can be considered noise. Another tradition relies on statistical nature of vast quantity of data, where geometric regularity is intrinsic to data and statistical models usually grasp this level only indirectly. This work focuses on point cloud data of natural resources and the silhouette recognition from video input as two real world examples of problems having geometric content which is intangible at the raw data presentation. This content could be discovered and modeled to some degree by such machine learning (ML) approaches like deep learning, but either a direct coverage of geometry in samples or addition of special geometry invariant layer is necessary. Geometric content is central when there is a need for direct observations of spatial variables, or one needs to gain a mapping to a geometrically consistent data representation, where e.g. outliers or noise can be easily discerned. In this thesis we consider transformation of original input data to a geometric feature space in two example problems. The first example is curvature of surfaces, which has met renewed interest since the introduction of ubiquitous point cloud data and the maturation of the discrete differential geometry. Curvature spectra can characterize a spatial sample rather well, and provide useful features for ML purposes. The second example involves projective methods used to video stereo-signal analysis in swimming analytics. The aim is to find meaningful local geometric representations for feature generation, which also facilitate additional analysis based on geometric understanding of the model. The features are associated directly to some geometric quantity, and this makes it easier to express the geometric constraints in a natural way, as shown in the thesis. Also, the visualization and further feature generation is much easier. Third, the approach provides sound baseline methods to more traditional ML approaches, e.g. neural network methods. Fourth, most of the ML methods can utilize the geometric features presented in this work as additional features.Geometriassa käytetään perinteisesti tarkkoja malleja, jolloin datassa esiintyvät epätarkkuudet edustavat melua. Toisessa perinteessä nojataan suuren datamäärän tilastolliseen luonteeseen, jolloin geometrinen säännönmukaisuus on datan sisäsyntyinen ominaisuus, joka hahmotetaan tilastollisilla malleilla ainoastaan epäsuorasti. Tämä työ keskittyy kahteen esimerkkiin: luonnonvaroja kuvaaviin pistepilviin ja videohahmontunnistukseen. Nämä ovat todellisia ongelmia, joissa geometrinen sisältö on tavoittamattomissa raakadatan tasolla. Tämä sisältö voitaisiin jossain määrin löytää ja mallintaa koneoppimisen keinoin, esim. syväoppimisen avulla, mutta joko geometria pitää kattaa suoraan näytteistämällä tai tarvitaan neuronien lisäkerros geometrisia invariansseja varten. Geometrinen sisältö on keskeinen, kun tarvitaan suoraa avaruudellisten suureiden havainnointia, tai kun tarvitaan kuvaus geometrisesti yhtenäiseen dataesitykseen, jossa poikkeavat näytteet tai melu voidaan helposti erottaa. Tässä työssä tarkastellaan datan muuntamista geometriseen piirreavaruuteen kahden esimerkkiohjelman suhteen. Ensimmäinen esimerkki on pintakaarevuus, joka on uudelleen virinneen kiinnostuksen kohde kaikkialle saatavissa olevan datan ja diskreetin geometrian kypsymisen takia. Kaarevuusspektrit voivat luonnehtia avaruudellista kohdetta melko hyvin ja tarjota koneoppimisessa hyödyllisiä piirteitä. Toinen esimerkki koskee projektiivisia menetelmiä käytettäessä stereovideosignaalia uinnin analytiikkaan. Tavoite on löytää merkityksellisiä paikallisen geometrian esityksiä, jotka samalla mahdollistavat muun geometrian ymmärrykseen perustuvan analyysin. Piirteet liittyvät suoraan johonkin geometriseen suureeseen, ja tämä helpottaa luonnollisella tavalla geometristen rajoitteiden käsittelyä, kuten väitöstyössä osoitetaan. Myös visualisointi ja lisäpiirteiden luonti muuttuu helpommaksi. Kolmanneksi, lähestymistapa suo selkeän vertailumenetelmän perinteisemmille koneoppimisen lähestymistavoille, esim. hermoverkkomenetelmille. Neljänneksi, useimmat koneoppimismenetelmät voivat hyödyntää tässä työssä esitettyjä geometrisia piirteitä lisäämällä ne muiden piirteiden joukkoon