We present a new family of linear time algorithms based on sufficient statistics for string comparison with mismatches under the string kernels framework. Our algorithms improve theoretical complexity bounds of existing approaches while scaling well with respect to the sequence alphabet size, the number of allowed mismatches and the size of the dataset. In particular, on large alphabets with loose mismatch constraints our algorithms are several orders of magnitude faster than the existing algorithms for string comparison under the mismatch similarity measure. We evaluate our algorithms on synthetic data and real applications in music genre classification, protein remote homology detection and protein fold prediction. The scalability of the algorithms allows us to consider complex sequence transformations, modeled using longer string features and larger numbers of mismatches, leading to a state-of-the-art performance with significantly reduced running times.

Learning directed acyclic graphs (DAGs) is long known a critical challenge at the core of probabilistic and causal modeling. The NoTears approach of (Zheng et al., 2018), through a differentiable function involving the matrix exponential trace $\mathrm{tr}(\exp(\cdot))$, opens up a way to learning DAGs via continuous optimization, though with a $O(d^3)$ complexity in the number $d$ of nodes. This paper presents a low-complexity model, called LoRAM for Low-Rank Additive Model, which combines low-rank matrix factorization with a sparsification mechanism for the continuous optimization of DAGs. The main contribution of the approach lies in an efficient gradient approximation method leveraging the low-rank property of the model, and its straightforward application to the computation of projections from graph matrices onto the DAG matrix space. The proposed method achieves a reduction from a cubic complexity to quadratic complexity while handling the same DAG characteristic function as NoTears, and scales easily up to thousands of nodes for the projection problem. The experiments show that the LoRAM achieves efficiency gains of orders of magnitude compared to the state-of-the-art at the expense of a very moderate accuracy loss in the considered range of sparse matrices, and with a low sensitivity to the rank choice of the model's low-rank component.

Linear temporal logic (LTL) is a specification language for finite sequences (called traces) widely used in program verification, motion planning in robotics, process mining, and many other areas. We consider the problem of learning LTL formulas for classifying traces; despite a growing interest of the research community, existing solutions suffer from two limitations: they do not scale beyond small formulas, and they may exhaust computational resources without returning any result. We introduce a new algorithm addressing both issues: our algorithm is able to construct formulas an order of magnitude larger than previous methods, and it is anytime, meaning that it in most cases successfully outputs a formula, albeit possibly not of minimal size. We evaluate the performances of our algorithm using an open source implementation against publicly available benchmarks.

Normalizing flows define a probability distribution by an explicit invertible transformation $\boldsymbol{\mathbf{z}}=f(\boldsymbol{\mathbf{x}})$. In this work, we present implicit normalizing flows (ImpFlows), which generalize normalizing flows by allowing the mapping to be implicitly defined by the roots of an equation $F(\boldsymbol{\mathbf{z}}, \boldsymbol{\mathbf{x}})= \boldsymbol{\mathbf{0}}$. ImpFlows build on residual flows (ResFlows) with a proper balance between expressiveness and tractability. Through theoretical analysis, we show that the function space of ImpFlow is strictly richer than that of ResFlows. Furthermore, for any ResFlow with a fixed number of blocks, there exists some function that ResFlow has a non-negligible approximation error. However, the function is exactly representable by a single-block ImpFlow. We propose a scalable algorithm to train and draw samples from ImpFlows. Empirically, we evaluate ImpFlow on several classification and density modeling tasks, and ImpFlow outperforms ResFlow with a comparable amount of parameters on all the benchmarks.

We present a new family of linear time algorithms based on sufficient statistics for string comparison with mismatches under the string kernels framework. Our algorithms improve theoretical complexity bounds of existing approaches while scaling well with respect to the sequence alphabet size, the number of allowed mismatches and the size of the dataset. In particular, on large alphabets with loose mismatch constraints our algorithms are several orders of magnitude faster than the existing algorithms for string comparison under the mismatch similarity measure. We evaluate our algorithms on synthetic data and real applications in music genre classification, protein remote homology detection and protein fold prediction. The scalability of the algorithms allows us to consider complex sequence transformations, modeled using longer string features and larger numbers of mismatches, leading to a state-of-the-art performance with significantly reduced running times.

In the Big Data era, many scientific and engineering domains are producing massive data streams, with petabyte and exabyte scales becoming increasingly common. Besides the explosive growth in volume, Big Data also has high velocity, high variety, and high uncertainty. These complex data streams require ever-increasing processing speeds, economical storage, and timely response for decision making in highly uncertain environments, and have raised various challenges to conventional data analysis. With the primary goal of building intelligent systems that automatically improve from experiences, machine learning (ML) is becoming an increasingly important field to tackle big data challenges, with an emerging field of "Big Learning," which covers theories, algorithms and systems on addressing big data problems. Bayesian methods have been widely used in machine learning and many other areas.

Recovering matrices from compressive and grossly corrupted observations is a fundamental problem in robust statistics, with rich applications in computer vision and machine learning. In theory, under certain conditions, this problem can be solved in polynomial time via a natural convex relaxation, known as Compressive Principal Component Pursuit (CPCP). However, all existing provable algorithms for CPCP suffer from superlinear per-iteration cost, which severely limits their applicability to large scale problems. In this paper, we propose provable, scalable and efficient methods to solve CPCP with (essentially) linear per-iteration cost. Our method combines classical ideas from Frank-Wolfe and proximal methods. In each iteration, we mainly exploit Frank-Wolfe to update the low-rank component with rank-one SVD and exploit the proximal step for the sparse term. Convergence results and implementation details are also discussed. We demonstrate the scalability of the proposed approach with promising numerical experiments on visual data.