Provable Dynamic Robust PCA or Robust Subspace Tracking

Dynamic robust PCA refers to the dynamic (time-varying) extension of robust PCA (RPCA). It assumes that the true (uncorrupted) data lies in a low-dimensional subspace that can change with time, albeit slowly. The goal is to track this changing subspace over time in the presence of sparse outliers. We develop and study a novel algorithm, that we call simple-ReProCS, based on the recently introduced Recursive Projected Compressive Sensing (ReProCS) framework. Our work provides the first guarantee for dynamic RPCA that holds under weakened versions of standard RPCA assumptions, slow subspace change and a lower bound assumption on most outlier magnitudes. Our result is significant because (i) it removes the strong assumptions needed by the two previous complete guarantees for ReProCS-based algorithms; (ii) it shows that it is possible to achieve significantly improved outlier tolerance, compared with all existing RPCA or dynamic RPCA solutions by exploiting the above two simple extra assumptions; and (iii) it proves that simple-ReProCS is online (after initialization), fast, and, has near-optimal memory complexity.Comment: Minor writing edits. The paper has been accepted to IEEE Transactions on Information Theor

Farmer response to rationed and uncertain irrigation supplies

Water resource management / Water use efficiency / Evapotranspiration / Agricultural production / Irrigated farming / Irrigation scheduling / Water allocation / Water supply / Water scarcity / Water delivery / Reservoirs / Uncertainty / Yield

Robust Subspace Learning: Robust PCA, Robust Subspace Tracking, and Robust Subspace Recovery

PCA is one of the most widely used dimension reduction techniques. A related easier problem is "subspace learning" or "subspace estimation". Given relatively clean data, both are easily solved via singular value decomposition (SVD). The problem of subspace learning or PCA in the presence of outliers is called robust subspace learning or robust PCA (RPCA). For long data sequences, if one tries to use a single lower dimensional subspace to represent the data, the required subspace dimension may end up being quite large. For such data, a better model is to assume that it lies in a low-dimensional subspace that can change over time, albeit gradually. The problem of tracking such data (and the subspaces) while being robust to outliers is called robust subspace tracking (RST). This article provides a magazine-style overview of the entire field of robust subspace learning and tracking. In particular solutions for three problems are discussed in detail: RPCA via sparse+low-rank matrix decomposition (S+LR), RST via S+LR, and "robust subspace recovery (RSR)". RSR assumes that an entire data vector is either an outlier or an inlier. The S+LR formulation instead assumes that outliers occur on only a few data vector indices and hence are well modeled as sparse corruptions.Comment: To appear, IEEE Signal Processing Magazine, July 201

Interfacial stresses and debonding failures in plated beams

Extensive research and recent developments in structural engineering has shown that adhesive bonding of fibre-reinforced polymer (FRP) composite, steel or any other metallic plate to the tension face of a reinforced concrete (RC), metallic or timber beam can effectively enhance its strength and other aspects of structural performance. This technique is now popularly adopted for retro-fitment and rehabilitation of existing structures. These plated beams often fail prematurely well before attaining the full flexural capacity by either plate end debonding (PED) or intermediate crack-induced interfacial debonding (ICD) failure. Concentration of higher interfacial shear and normal stresses at the plate end due to a geometric discontinuity is believed to be responsible for PED that initiates at the plate end and propagates inwards. PED includes concrete cover separation and interfacial debonding initiated at the plate end; and such failure initiated at a critical diagonal crack. ICD initiates at an intermediate major flexural or flexural-shear crack in the soffit of the original beam due to high bond stress and propagates towards one of the plate ends (type-1) or an adjacent crack (type-2). This thesis presents a study of interfacial stresses and debonding failures in plated beams. It first presents a simple and novel theoretical solution of interfacial stresses applicable to any loading considering major deformations like axial and flexural deformations in the beam and plate within linear elastic range. This solution is then enhanced with the inclusion of the effect of adherends’ shear deformation by approximating the displacement field for interfacial shear stress and using Timoshenko’s beam theory for interfacial normal stress, achieving a better understanding of the effect of shear deformation which is ill-understood. This resulted in a first ever solution to include the effect of adherends’ shear deformation under both interfacial shear and normal stresses. This solution is further advanced by developing a rigorous and a versatile closed-form solution fully based on Timoshenko’s beam theory that offered a significant insight. Interfacial stresses at the plate end cannot be measured directly using available measurement techniques, and may only be interpreted indirectly from measured plate strains. The conventional interpretation is based on the assumption that the plate is under pure tension. A significant drawback of this is that the interfacial normal stresses iii cannot be deduced. A new technique is developed to deduce both interfacial shear and normal stresses from strain measurements. The thesis presents three PED strength models for the special case of an RC beam with the plate terminated in the constant moment region: a theoretical model based on interfacial fracture mechanics with a reasonable accuracy; a semi-empirical model with greater accuracy; and an empirical model that is slightly less accurate but simpler to apply than the semi-empirical model. This is followed by the development of a shear debonding model to predict the debonding failure in an RC beam with the plate terminated in high shear and a very low or zero moment region. The two models for PED failure in pure bending and pure shear zones are then combined to result in an accurate shear-bending interaction debonding model. An assessment of these models against a carefully constructed large test database shows that they are more accurate than existing models and suitable for implementation in design codes or guidelines. Finally, a structural mechanics formulation for an FRP-to-concrete bonded joint between two adjacent cracks is developed. It considers axial forces, transverse shear forces and bending moments in the adherends and uses a linearly softening bond-slip model. A section analysis with partial interaction and a rotational spring method are used to relate the applied loading to the interfacial deformation. A closed-form solution is obtained that may form the basis of a rational ICD design method

Fast Robust Subspace Tracking via PCA in Sparse Data-Dependent Noise

This work studies the robust subspace tracking (ST) problem. Robust ST can be simply understood as a (slow) time-varying subspace extension of robust PCA. It assumes that the true data lies in a low-dimensional subspace that is either fixed or changes slowly with time. The goal is to track the changing subspaces over time in the presence of additive sparse outliers and to do this quickly (with a short delay). We introduce a "fast" mini-batch robust ST solution that is provably correct under mild assumptions. Here "fast" means two things: (i) the subspace changes can be detected and the subspaces can be tracked with near-optimal delay, and (ii) the time complexity of doing this is the same as that of simple (non-robust) PCA. Our main result assumes piecewise constant subspaces (needed for identifiability), but we also provide a corollary for the case when there is a little change at each time. A second contribution is a novel non-asymptotic guarantee for PCA in linearly data-dependent noise. An important setting where this is useful is for linearly data dependent noise that is sparse with support that changes enough over time. The analysis of the subspace update step of our proposed robust ST solution uses this result.Comment: To appear in IEEE Journal of Special Areas in Information Theor