Under-counted matrix completion (UC-MC) has many important applications, especially in epidemiology and ecology where the observed data are often smaller than the actual numbers. Existing works model the under-counting effects using entry-wise miss detection probabilities, which are usually formulated as functions of detection-related side information or features (e.g., weather conditions for observing a certain species). However, such features are not always available. This work proposes a model for UC-MC that circumvents using such side information. By assuming that the detection probabilities for a large proportion of entries are similar, the under-counting probabilities are approximated by a specially structured (i.e., rank-one plus sparse) matrix. This way, a structure-regularized UC-MC formulation is attained, and no detection-related side information is used. A re-parameterization-based implementation is proposed, allowing us to employ off-the-shelf gradient-based optimizers to tackle the loss function of interest. Simulations are used to demonstrate the effectiveness of the proposed approach.

The generation of label noise is often modeled as a process involving a probability transition matrix (also interpreted as the _annotator confusion matrix_) imposed onto the label distribution. Under this model, learning the ``ground-truth classifier''---i.e., the classifier that can be learned if no noise was present---and the confusion matrix boils down to a model identification problem. Prior works along this line demonstrated appealing empirical performance, yet identifiability of the model was mostly established by assuming an instance-invariant confusion matrix. Having an (occasionally) instance-dependent confusion matrix across data samples is apparently more realistic, but inevitably introduces outliers to the model. Our interest lies in confusion matrix-based noisy label learning with such outliers taken into consideration. We begin with pointing out that under the model of interest, using labels produced by only one annotator is fundamentally insufficient to detect the outliers or identify the ground-truth classifier. Then, we prove that by employing a crowdsourcing strategy involving multiple annotators, a carefully designed loss function can establish the desired model identifiability under reasonable conditions. Our development builds upon a link between the noisy label model and a column-corrupted matrix factorization mode---based on which we show that crowdsourced annotations distinguish nominal data and instance-dependent outliers using a low-dimensional subspace. Experiments show that our learning scheme substantially improves outlier detection and the classifier's testing accuracy.

The recent integration of deep learning and pairwise similarity annotation-based constrained clustering—i.e., deep constrained clustering (DCC)—has proven effective for incorporating weak supervision into massive data clustering: Less than 1% of pair similarity annotations can often substantially enhance the clustering accuracy. However, beyond empirical successes, there is a lack of understanding of DCC. In addition, many DCC paradigms are sensitive to annotation noise, but performance-guaranteed noisy DCC methods have been largely elusive. This work first takes a deep look into a recently emerged logistic loss function of DCC, and characterizes its theoretical properties. Our result shows that the logistic DCC loss ensures the identifiability of data membership under reasonable conditions, which may shed light on its effectiveness in practice. Building upon this understanding, a new loss function based on geometric factor analysis is proposed to fend against noisy annotations. It is shown that even under unknown annotation confusions, the data membership can still be provably identified under our proposed learning criterion. The proposed approach is tested over multiple datasets to validate our claims.