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Abstract Sea surface height observations provided by satellite altimetry since 1993 show a rising rate (3.4 mm yr⁻¹) for global mean sea level. While on average, sea level has risen 10 cm over the last 30 years, there is considerable regional variation in the sea level change. Through this work, we predict sea level trends 30 years into the future at a 2° spatial resolution and investigate the future patterns of the sea level change. We show the potential of machine learning (ML) in this challenging application of long-term sea level forecasting over the global ocean. Our approach incorporates sea level data from both altimeter observations and climate model simulations. We develop a supervised learning framework using fully connected neural networks (FCNNs) that can predict the sea level trend based on climate model projections. Alongside this, our method provides uncertainty estimates associated with the ML prediction. We also show the effectiveness of partitioning our spatial dataset and learning a dedicated ML model for each segmented region. We compare two partitioning strategies: one achieved using domain knowledge and the other employing spectral clustering. Our results demonstrate that segmenting the spatial dataset with spectral clustering improves the ML predictions. Significance StatementLong-term projections are needed to help coastal communities adapt to sea level rise. Forecasting multidecadal sea level change is a complex problem. In this paper, we show the promise of machine learning in producing such forecasts 30 years in advance and over the global ocean. Continued improvements in prediction skills that build on this work will be vital in sea level rise adaptation efforts. 

We build upon recent work on the use of machine-learning models to estimate Hamiltonian parameters using continuous weak measurement of qubits as input. We consider two settings for the training of our model: (1) supervised learning, where the weak-measurement training record can be labeled with known Hamiltonian parameters, and (2) unsupervised learning, where no labels are available. The first has the advantage of not requiring an explicit representation of the quantum state, thus potentially scaling very favorably to a larger number of qubits. The second requires the implementation of a physical model to map the Hamiltonian parameters to a measurement record, which we implement using an integrator of the physical model with a recurrent neural network to provide a model-free correction at every time step to account for small effects not captured by the physical model. We test our construction on a system of two qubits and demonstrate accurate prediction of multiple physical parameters in both the supervised context and the unsupervised context. We demonstrate that the model benefits from larger training sets, establishing that it is “learning,” and we show robustness regarding errors in the assumed physical model by achieving accurate parameter estimation in the presence of unanticipated single-particle relaxation. 

The Nyström method is a matrix approximation technique that has shown great promise in speeding up spectral clustering. However, when the input matrix is sparse, we show that the traditional Nyström method requires a prohibitively large number of samples to obtain a good approximation. We propose a novel sampling approach to select the landmark points used to compute the Nyström approximation. We show that the proposed sampling approach obeys the same error bound as in Bouneffouf and Birol (2015). To control sample complexity, we propose a selective densification step based on breadth-first traversal. We show that the proposed densification does not change the optimal clustering. Results on real world datasets show that by combining the proposed sampling and densification schemes, we can obtain better accuracy compared to other techniques used for the Nyström method while using significantly fewer samples. 

We analyze online (Bottou & Bengio, 1994) and mini-batch (Sculley, 2010) k-means variants. Both scale up the widely used Lloyd’s algorithm via stochastic approximation, and have become popular for large-scale clustering and unsupervised feature learning. We show, for the first time, that they have global convergence towards “local optima” at rate O(1/t) under general conditions. In addition, we show that if the dataset is clusterable, stochastic k-means with suitable initialization converges to an optimal k-means solution at rate O(1/t) with high probability. The k-means objective is non-convex and non-differentiable; we exploit ideas from non-convex gradient-based optimization by providing a novel characterization of the trajectory of the k-means algorithm on its solution space, and circumvent its non-differentiability via geometric insights about the k-means update.