We will incorporate these experiments into the final paper.

Implicit regularization is an important way to interpret neural networks. Recent theory starts to explain implicit regularization with the model of deep matrix factorization (DMF) and analyze the trajectory of discrete gradient dynamics in the optimization process. These discrete gradient dynamics are relatively small but not infinitesimal, thus fitting well with the practical implementation of neural networks. Currently, discrete gradient dynamics analysis has been successfully applied to shallow networks but encounters the difficulty of complex computation for deep networks. In this work, we introduce another discrete gradient dynamics approach to explain implicit regularization, i.e. landscape analysis. It mainly focuses on gradient regions, such as saddle points and local minima. We theoretically establish the connection between saddle point escaping (SPE) stages and the matrix rank in DMF. We prove that, for a rank-R matrix reconstruction, DMF will converge to a second-order critical point after R stages of SPE. This conclusion is further experimentally verified on a low-rank matrix reconstruction problem. This work provides a new theory to analyze implicit regularization in deep learning.

With rapid progress in deep learning, neural networks have been widely used in scientific research and engineering applications as surrogate models. Despite the great success of neural networks in fitting complex systems, two major challenges still remain: i) the lack of generalization on different problems/datasets, and ii) the demand for large amounts of simulation data that are computationally expensive. To resolve these challenges, we propose the differentiable \mf (DMF) model, which leverages neural architecture search (NAS) to automatically search the suitable model architecture for different problems, and transfer learning to transfer the learned knowledge from low-fidelity (fast but inaccurate) data to high-fidelity (slow but accurate) model. Novel and latest machine learning techniques such as hyperparameters search and alternate learning are used to improve the efficiency and robustness of DMF. As a result, DMF can efficiently learn the physics simulations with only a few high-fidelity training samples, and outperform the state-of-the-art methods with a significant margin (with up to 58$\%$ improvement in RMSE) based on a variety of synthetic and practical benchmark problems.

Digital microfluidics (DMF) has the signatures of an ideal liquid handling platform – as shown through almost two decades of automated biological and chemical assays. However, in the current state of DMF, we are still limited by the number of parallel biological or chemical assays that can be performed on DMF. Here, we report a new approach that leverages design-of-experiment and numerical methodologies to accelerate experimental optimization on DMF. The integration of the one-factor-at-a-time (OFAT) experimental technique with machine learning algorithms provides a set of recommended optimal conditions without the need to perform a large set of experiments. We applied our approach towards optimizing the radiochemistry synthesis yield given the large number of variables that affect the yield. We believe that this work is the first to combine such techniques which can be readily applied to any other assays that contain many parameters and levels on DMF.

The explicit low-rank regularization, e.g., nuclear norm regularization, has been widely used in imaging sciences. However, it has been found that implicit regularization outperforms explicit ones in various image processing tasks. Another issue is that the fixed explicit regularization limits the applicability to broad images since different images favor different features captured by different explicit regularizations. As such, this paper proposes a new adaptive and implicit low-rank regularization that captures the low-rank prior dynamically from the training data. The core of our new adaptive and implicit low-rank regularization is parameterizing the Laplacian matrix in the Dirichlet energy-based regularization, which we call the regularization AIR. Theoretically, we show that the adaptive regularization of \ReTwo{AIR} enhances the implicit regularization and vanishes at the end of training. We validate AIR's effectiveness on various benchmark tasks, indicating that the AIR is particularly favorable for the scenarios when the missing entries are non-uniform. The code can be found at https://github.com/lizhemin15/AIR-Net.

Variational auto-encoders (VAE) are popular deep latent variable models which are trained by maximizing an Evidence Lower Bound (ELBO). To obtain tighter ELBO and hence better variational approximations, it has been proposed to use importance sampling to get a lower variance estimate of the evidence. However, importance sampling is known to perform poorly in high dimensions. While it has been suggested many times in the literature to use more sophisticated algorithms such as Annealed Importance Sampling (AIS) and its Sequential Importance Sampling (SIS) extensions, the potential benefits brought by these advanced techniques have never been realized for VAE: the AIS estimate cannot be easily differentiated, while SIS requires the specification of carefully chosen backward Markov kernels. In this paper, we address both issues and demonstrate the performance of the resulting Monte Carlo VAEs on a variety of applications.

Points of interest (POI) recommendation has been drawn much attention recently due to the increasing popularity of location-based networks, e.g., Foursquare and Yelp. Among the existing approaches to POI recommendation, Matrix Factorization (MF) based techniques have proven to be effective. However, existing MF approaches suffer from two major problems: (1) Expensive computations and storages due to the centralized model training mechanism: the centralized learners have to maintain the whole user-item rating matrix, and potentially huge low rank matrices. (2) Privacy issues: the users' preferences are at risk of leaking to malicious attackers via the centralized learner. To solve these, we present a Decentralized MF (DMF) framework for POI recommendation. Specifically, instead of maintaining all the low rank matrices and sensitive rating data for training, we propose a random walk based decentralized training technique to train MF models on each user's end, e.g., cell phone and Pad. By doing so, the ratings of each user are still kept on one's own hand, and moreover, decentralized learning can be taken as distributed learning with multi-learners (users), and thus alleviates the computation and storage issue. Experimental results on two real-world datasets demonstrate that, comparing with the classic and state-of-the-art latent factor models, DMF significantly improvements the recommendation performance in terms of precision and recall.

We address the problem of reconstructing a matrix from a subset of its entries. Current methods, branded as geometric matrix completion, augment classical rank regularization techniques by incorporating geometric information into the solution. This information is usually provided as graphs encoding relations between rows/columns. In this work we propose a simple spectral approach for solving the matrix completion problem, via the framework of functional maps. We introduce the zoomout loss, a multiresolution spectral geometric loss inspired by recent advances in shape correspondence, whose minimization leads to state-of-the-art results on various recommender systems datasets. Surprisingly, for some datasets we were able to achieve comparable results even without incorporating geometric information. This puts into question both the quality of such information and current methods' ability to use it in a meaningful and efficient way.

To adhere to these laws, insurers either built or outsourced processes to match their policyholder data to the DMF or similar databases; however, the match criteria outlined in state regulatory agreements necessitated a secondary death validation to confirm a policyholder death – often a manual and time-consuming process. To complicate matters, thousands of erroneous deaths are reported in the DMF each year, further necessitating a sound validation process that mitigates the risk of a falsely reported death from a single source. A death validation process that cross-references deaths reported in the DMF against other death sources, such as state vital statistics and obituaries, can reduce or eliminate the manual process that insurance companies use to validate deaths. This streamlined process can be automated for millions of records and completed in a fraction of the time that it would require of a person or team. It should use common data elements available across multiple death sources, such as name and death date, to identify the same decedent, but also accommodate for slight variations (e.g., nicknames, misspellings) and false positive reduction (e.g., common names) that might otherwise result in no matches or mismatches.