Many problems in signal processing, machine learning and computer vision can be solved by learning low rank models from data. In computer vision, problems such as rigid structure from motion have been formulated as an optimization over subspaces with fixed rank. These hard-rank constraints have traditionally been imposed by a factorization that parameterizes subspaces as a product of two matrices of fixed rank. Whilst factorization approaches lead to efficient and kernelizable optimization algorithms, they have been shown to be NP-Hard in presence of missing data. Inspired by recent work in compressed sensing, hard-rank constraints have been replaced by soft-rank constraints, such as the nuclear norm regularizer. Vis-a-vis hard-rank approaches, soft-rank models are convex even in presence of missing data: but how is convex optimization solving a NP-Hard problem? This thesis addresses this question by analyzing the relationship between hard and soft rank constraints in the unsupervised factorization with missing data problem. Moreover, we extend soft rank models to weakly supervised and fully supervised learning problems in computer vision. There are four main contributions of our work: (1) The analysis of a new unified low-rank model for matrix factorization with missing data. Our model subsumes soft and hard-rank approaches and merges advantages from previous formulations, such as efficient algorithms and kernelization. It also provides justifications on the choice of algorithms and regions that guarantee convergence to global minima. (2) A deterministic \rank continuation" strategy for the NP-hard unsupervised factorization with missing data problem, that is highly competitive with the state-of-the-art and often achieves globally optimal solutions. In preliminary work, we show that this optimization strategy is applicable to other NP-hard problems which are typically relaxed to convex semidentite programs (e.g., MAX-CUT, quadratic assignment problem). (3) A new soft-rank fully supervised robust regression model. This convex model is able to deal with noise, outliers and missing data in the input variables. (4) A new soft-rank model for weakly supervised image classification and localization. Unlike existing multiple-instance approaches for this problem, our model is convex.

High levels of polychlorinated dibenzo-p-dioxins and polychlorinated dibenzofurans (PCDD/Fs) in edible fish in the Baltic Sea have raised health concerns in the Baltic region and the rest of Europe. Thus, there are urgent needs to characterize sources in order to formulate effective mitigation strategies. The aim of this thesis is to contribute to a better understanding of past and present sources of PCDD/Fs in the Baltic Sea environment by exploring chemical fingerprints in sediments, air, and biota. The spatial and temporal patterns of PCDD/F distributions in the Baltic Sea during the 20th century were studied in Swedish coastal and offshore sediment cores. The results showed that PCDD/F levels peaked in 1975 (± 7 years) in coastal and 1991 (± 5 years) in offshore areas. The time trends of PCDD/Fs in the sediment cores also showed that environmental half-lives of these pollutants have been shorter in coastal than in offshore areas (15 ± 5 and 29 ± 14 years, respectively). Consequently, there have been remarkable recoveries in coastal areas, but slower recovery in offshore areas with 81 ± 12% and 38 ± 11% reductions from peak levels, respectively. Source-to-receptor multivariate modeling by Positive Matrix Factorization (PMF) showed that six types of PCDD/F sources are and have been important for the Baltic Sea environment: PCDD/Fs related to i) atmospheric background, ii) thermal processes, iii) manufacture and use of tetra-chlorophenol (TCP) and iv) penta-chlorophenol (PCP), v) industrial use of elementary chlo- rine and the chloralkali-process (Chl), and vi) hexa-CDD sources. The results showed that diffuse sources (i and ii) have consistently contributed >80% of the total amounts in the Southern Baltic Sea. In the Northern Baltic Sea, where the biota is most heavily contaminated, impacts of local sources (TCP, PCP and Chl) have been higher, contributing ca. 50% of total amounts. Among the six sources, only Thermal and chlorophenols (ii-iv) have had major impacts on biota. The impact of thermal sources has, however, been declining as shown from source apportioned time-trend data of PCDD/Fs in Baltic herring. In contrast, impacts of chlorophenol-associated sources generally increased, remained at steady-state or slowly decreased during 1990-2010, suggesting that these sources have substantially contributed to the persistently high levels of PCDD/Fs in Baltic biota. Atmospheric sources of PCDD/Fs for the Baltic region (Northern Europe) were also investigated, and specifically whether the inclusion of parallel measurements of metals in the analysis of air would help back-tracking sources. PCDD/Fs and metals in high-volume air samples from a rural field station near the shore of the central Baltic Sea were measured. The study focused on the winter season and air from the S and E sectors, as these samples showed elevated levels of PCDD/Fs, particularly PCDFs. Several metals were found to correlate significantly with the PCDFs. The wide range of candidate metals as source markers for PCDD/F emissions, and the lack of an up-to-date extensive compilation of source characteristics for metal emission from vari- ous sources, limited the use of the metals as source markers. The study was not able to pin-point primary PCDD/F sources for Baltic air, but it demonstrated a new promising approach for source tracing of air emissions. The best leads for back-tracking primary sources of atmospheric PCDD/Fs in Baltic air were seasonal trends and PCDD/F congener patterns, pointing at non-industrial related thermal sources related to heating. The non-localized natures of the sources raise challenges for managing the emissions and thus societal efforts are required to better control atmospheric emissions of PCDD/Fs. / EcoChange / BalticPOPs