Contrastive dimension reduction methods have been developed for case-control study data to identify variation that is enriched in the foreground (case) data X relative to the background (control) data Y. Here, we develop contrastive regression for the setting when there is a response variable r associated with each foreground observation. This situation occurs frequently when, for example, the unaffected controls do not have a disease grade or intervention dosage but the affected cases have a disease grade or intervention dosage, as in autism severity, solid tumors stages, polyp sizes, or warfarin dosages. Our contrastive regression model captures shared low-dimensional variation between the predictors in the cases and control groups, and then explains the case-specific response variables through the variance that remains in the predictors after shared variation is removed. We show that, in one single-nucleus RNA sequencing dataset on autism severity in postmortem brain samples from donors with and without autism and in another single-cell RNA sequencing dataset on cellular differentiation in chronic rhinosinusitis with and without nasal polyps, our contrastive linear regression performs feature ranking and identifies biologically-informative predictors associated with response that cannot be identified using other approaches
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Rapid progress in scalable, commoditized tools for data collection and data processing has made it possible for firms and policymakers to employ ever more complex metrics as guides for decision-making. These developments have highlighted a prevailing challenge -- deciding *which* metrics to compute. In particular, a firm's ability to compute a wider range of existing metrics does not address the problem of *unknown unknowns*, which reflects informational limitations on the part of the firm. To guide the choice of metrics in the face of this informational problem, we turn to the evaluated agents themselves, who may have more information than a principal about how to measure outcomes effectively. We model this interaction as a simple agency game, where we ask: *When does an agent have an incentive to reveal the observability of a cost-correlated variable to the principal?* There are two effects: better information reduces the agent's information rents but also makes some projects go forward that otherwise would fail. We show that the agent prefers to reveal information that exposes a strong enough differentiation between high and low costs. Expanding the agent's action space to include the ability to *garble* their information, we show that the agent often prefers to garble over full revelation. Still, giving the agent the ability to garble can lead to higher total welfare. Our model has analogies with price discrimination, and we leverage some of these synergies to analyze total welfare.
We consider finite element approximations of unique continuation problems subject to elliptic equations in the case where the normal derivative of the exact solution is known to reside in some finite dimensional space. To give quantitative error estimates we prove Lipschitz stability of the unique continuation problem in the global H1-norm. This stability is then leveraged to derive optimal a posteriori and a priori error estimates for a primal-dual stabilised finite method.
Group lasso is a commonly used regularization method in statistical learning in which parameters are eliminated from the model according to predefined groups. However, when the groups overlap, optimizing the group lasso penalized objective can be time-consuming on large-scale problems because of the non-separability induced by the overlapping groups. This bottleneck has seriously limited the application of overlapping group lasso regularization in many modern problems, such as gene pathway selection and graphical model estimation. In this paper, we propose a separable penalty as an approximation of the overlapping group lasso penalty. Thanks to the separability, the computation of regularization based on our penalty is substantially faster than that of the overlapping group lasso, especially for large-scale and high-dimensional problems. We show that the penalty is the tightest separable relaxation of the overlapping group lasso norm within the family of $\ell_{q_1}/\ell_{q_2}$ norms. Moreover, we show that the estimator based on the proposed separable penalty is statistically equivalent to the one based on the overlapping group lasso penalty with respect to their error bounds and the rate-optimal performance under the squared loss. We demonstrate the faster computational time and statistical equivalence of our method compared with the overlapping group lasso in simulation examples and a classification problem of cancer tumors based on gene expression and multiple gene pathways.
We study matching markets with aligned preferences and establish a connection between common design objectives -- stability, efficiency, and fairness -- and the theory of optimal transport. Optimal transport gives new insights into the structural properties of matchings obtained from pursuing these objectives, and into the trade-offs between different objectives. Matching markets with aligned preferences provide a tractable stylized model capturing supply-demand imbalances in a range of settings such as partnership formation, school choice, organ donor exchange, and markets with transferable utility where bargaining over transfers happens after a match is formed.
Images when processed using various enhancement techniques often lead to edge degradation and other unwanted artifacts such as halos. These artifacts pose a major problem for photographic applications where they can denude the quality of an image. There is a plethora of edge-aware techniques proposed in the field of image processing. However, these require the application of complex optimization or post-processing methods. Local Laplacian Filtering is an edge-aware image processing technique that involves the construction of simple Gaussian and Laplacian pyramids. This technique can be successfully applied for detail smoothing, detail enhancement, tone mapping and inverse tone mapping of an image while keeping it artifact-free. The problem though with this approach is that it is computationally expensive. Hence, parallelization schemes using multi-core CPUs and GPUs have been proposed. As is well known, they are not power-efficient, and a well-designed hardware architecture on an FPGA can do better on the performance per watt metric. In this paper, we propose a hardware accelerator, which exploits fully the available parallelism in the Local Laplacian Filtering algorithm, while minimizing the utilization of on-chip FPGA resources. On Virtex-7 FPGA, we obtain a 7.5x speed-up to process a 1 MB image when compared to an optimized baseline CPU implementation. To the best of our knowledge, we are not aware of any other hardware accelerators proposed in the research literature for the Local Laplacian Filtering problem.
Marginal structural models have been widely used in causal inference to estimate mean outcomes under either a static or a prespecified set of treatment decision rules. This approach requires imposing a working model for the mean outcome given a sequence of treatments and possibly baseline covariates. In this paper, we introduce a dynamic marginal structural model that can be used to estimate an optimal decision rule within a class of parametric rules. Specifically, we will estimate the mean outcome as a function of the parameters in the class of decision rules, referred to as a regimen-response curve. In general, misspecification of the working model may lead to a biased estimate with questionable causal interpretability. To mitigate this issue, we will leverage risk to assess "goodness-of-fit" of the imposed working model. We consider the counterfactual risk as our target parameter and derive inverse probability weighting and canonical gradients to map it to the observed data. We provide asymptotic properties of the resulting risk estimators, considering both fixed and data-dependent target parameters. We will show that the inverse probability weighting estimator can be efficient and asymptotic linear when the weight functions are estimated using a sieve-based estimator. The proposed method is implemented on the LS1 study to estimate a regimen-response curve for patients with Parkinson's disease.
The splitting-off operation in undirected graphs is a fundamental reduction operation that detaches all edges incident to a given vertex and adds new edges between the neighbors of that vertex while preserving their degrees. Lov\'asz (1974) and Mader (1978) showed the existence of this operation while preserving global and local connectivities respectively in graphs under certain conditions. These results have far-reaching applications in graph algorithms literature. In this work, we introduce a splitting-off operation in hypergraphs. We show that there exists a local connectivity preserving complete splitting-off in hypergraphs and give a strongly polynomial-time algorithm to compute it in weighted hypergraphs. We illustrate the usefulness of our splitting-off operation in hypergraphs by showing two applications: (1) we give a constructive characterization of $k$-hyperedge-connected hypergraphs and (2) we give an alternate proof of an approximate min-max relation for max Steiner rooted-connected orientation of graphs and hypergraphs (due to Kir\'aly and Lau (Journal of Combinatorial Theory, 2008; FOCS 2006)). Our proof of the approximate min-max relation for graphs circumvents the Nash-Williams' strong orientation theorem and uses tools developed for hypergraphs.
The adaptive processing of structured data is a long-standing research topic in machine learning that investigates how to automatically learn a mapping from a structured input to outputs of various nature. Recently, there has been an increasing interest in the adaptive processing of graphs, which led to the development of different neural network-based methodologies. In this thesis, we take a different route and develop a Bayesian Deep Learning framework for graph learning. The dissertation begins with a review of the principles over which most of the methods in the field are built, followed by a study on graph classification reproducibility issues. We then proceed to bridge the basic ideas of deep learning for graphs with the Bayesian world, by building our deep architectures in an incremental fashion. This framework allows us to consider graphs with discrete and continuous edge features, producing unsupervised embeddings rich enough to reach the state of the art on several classification tasks. Our approach is also amenable to a Bayesian nonparametric extension that automatizes the choice of almost all model's hyper-parameters. Two real-world applications demonstrate the efficacy of deep learning for graphs. The first concerns the prediction of information-theoretic quantities for molecular simulations with supervised neural models. After that, we exploit our Bayesian models to solve a malware-classification task while being robust to intra-procedural code obfuscation techniques. We conclude the dissertation with an attempt to blend the best of the neural and Bayesian worlds together. The resulting hybrid model is able to predict multimodal distributions conditioned on input graphs, with the consequent ability to model stochasticity and uncertainty better than most works. Overall, we aim to provide a Bayesian perspective into the articulated research field of deep learning for graphs.
Conventional entity typing approaches are based on independent classification paradigms, which make them difficult to recognize inter-dependent, long-tailed and fine-grained entity types. In this paper, we argue that the implicitly entailed extrinsic and intrinsic dependencies between labels can provide critical knowledge to tackle the above challenges. To this end, we propose \emph{Label Reasoning Network(LRN)}, which sequentially reasons fine-grained entity labels by discovering and exploiting label dependencies knowledge entailed in the data. Specifically, LRN utilizes an auto-regressive network to conduct deductive reasoning and a bipartite attribute graph to conduct inductive reasoning between labels, which can effectively model, learn and reason complex label dependencies in a sequence-to-set, end-to-end manner. Experiments show that LRN achieves the state-of-the-art performance on standard ultra fine-grained entity typing benchmarks, and can also resolve the long tail label problem effectively.
It is always well believed that modeling relationships between objects would be helpful for representing and eventually describing an image. Nevertheless, there has not been evidence in support of the idea on image description generation. In this paper, we introduce a new design to explore the connections between objects for image captioning under the umbrella of attention-based encoder-decoder framework. Specifically, we present Graph Convolutional Networks plus Long Short-Term Memory (dubbed as GCN-LSTM) architecture that novelly integrates both semantic and spatial object relationships into image encoder. Technically, we build graphs over the detected objects in an image based on their spatial and semantic connections. The representations of each region proposed on objects are then refined by leveraging graph structure through GCN. With the learnt region-level features, our GCN-LSTM capitalizes on LSTM-based captioning framework with attention mechanism for sentence generation. Extensive experiments are conducted on COCO image captioning dataset, and superior results are reported when comparing to state-of-the-art approaches. More remarkably, GCN-LSTM increases CIDEr-D performance from 120.1% to 128.7% on COCO testing set.
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