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In this paper a set of previous general results for the development of B--series for a broad class of stochastic differential equations has been collected. The applicability of these results is demonstrated by the derivation of B--series for non-autonomous semi-linear SDEs and exponential Runge-Kutta methods applied to this class of SDEs, which is a significant generalization of existing theory on such methods.

Gaussian boson sampling, a computational model that is widely believed to admit quantum supremacy, has already been experimentally demonstrated and is claimed to surpass the classical simulation capabilities of even the most powerful supercomputers today. However, whether the current approach limited by photon loss and noise in such experiments prescribes a scalable path to quantum advantage is an open question. To understand the effect of photon loss on the scalability of Gaussian boson sampling, we analytically derive the asymptotic operator entanglement entropy scaling, which relates to the simulation complexity. As a result, we observe that efficient tensor network simulations are likely possible under the $N_\text{out}\propto\sqrt{N}$ scaling of the number of surviving photons orange$N_\text{out}$ in the number of input photons $N$. We numerically verify this result using a tensor network algorithm with $U(1)$ symmetry, and overcome previous challenges due to the large local Hilbert space dimensions in Gaussian boson sampling with hardware acceleration. Additionally, we observe that increasing the photon number through larger squeezing does not increase the entanglement entropy significantly. Finally, we numerically find the bond dimension necessary for fixed accuracy simulations, providing more direct evidence for the complexity of tensor networks.

Spectral independence is a recently-developed framework for obtaining sharp bounds on the convergence time of the classical Glauber dynamics. This new framework has yielded optimal $O(n \log n)$ sampling algorithms on bounded-degree graphs for a large class of problems throughout the so-called uniqueness regime, including, for example, the problems of sampling independent sets, matchings, and Ising-model configurations. Our main contribution is to relax the bounded-degree assumption that has so far been important in establishing and applying spectral independence. Previous methods for avoiding degree bounds rely on using $L^p$-norms to analyse contraction on graphs with bounded connective constant (Sinclair, Srivastava, Yin; FOCS'13). The non-linearity of $L^p$-norms is an obstacle to applying these results to bound spectral independence. Our solution is to capture the $L^p$-analysis recursively by amortising over the subtrees of the recurrence used to analyse contraction. Our method generalises previous analyses that applied only to bounded-degree graphs. As a main application of our techniques, we consider the random graph $G(n,d/n)$, where the previously known algorithms run in time $n^{O(\log d)}$ or applied only to large $d$. We refine these algorithmic bounds significantly, and develop fast $n^{1+o(1)}$ algorithms based on Glauber dynamics that apply to all $d$, throughout the uniqueness regime.

A key challenge when trying to understand innovation is that it is a dynamic, ongoing process, which can be highly contingent on ephemeral factors such as culture, economics, or luck. This means that any analysis of the real-world process must necessarily be historical - and thus probably too late to be most useful - but also cannot be sure what the properties of the web of connections between innovations is or was. Here I try to address this by designing and generating a set of synthetic innovation web "dictionaries" that can be used to host sampled innovation timelines, probe the overall statistics and behaviours of these processes, and determine the degree of their reliance on the structure or generating algorithm. Thus, inspired by the work of Fink, Reeves, Palma and Farr (2017) on innovation in language, gastronomy, and technology, I study how new symbol discovery manifests itself in terms of additional "word" vocabulary being available from dictionaries generated from a finite number of symbols. Several distinct dictionary generation models are investigated using numerical simulation, with emphasis on the scaling of knowledge as dictionary generators and parameters are varied, and the role of which order the symbols are discovered in.

Hierarchical time series are common in several applied fields. The forecasts for these time series are required to be coherent, that is, to satisfy the constraints given by the hierarchy. The most popular technique to enforce coherence is called reconciliation, which adjusts the base forecasts computed for each time series. However, recent works on probabilistic reconciliation present several limitations. In this paper, we propose a new approach based on conditioning to reconcile any type of forecast distribution. We then introduce a new algorithm, called Bottom-Up Importance Sampling, to efficiently sample from the reconciled distribution. It can be used for any base forecast distribution: discrete, continuous, or in the form of samples, providing a major speedup compared to the current methods. Experiments on several temporal hierarchies show a significant improvement over base probabilistic forecasts.

Neural ordinary differential equations (neural ODEs) are a popular family of continuous-depth deep learning models. In this work, we consider a large family of parameterized ODEs with continuous-in-time parameters, which include time-dependent neural ODEs. We derive a generalization bound for this class by a Lipschitz-based argument. By leveraging the analogy between neural ODEs and deep residual networks, our approach yields in particular a generalization bound for a class of deep residual networks. The bound involves the magnitude of the difference between successive weight matrices. We illustrate numerically how this quantity affects the generalization capability of neural networks.

Deep Learning and transfer learning models are being used to generate time series forecasts; however, there is scarce evidence about their performance prediction that it is more evident for monthly time series. The purpose of this paper is to compare Deep Learning models with transfer learning and without transfer learning and other traditional methods used for monthly forecasts to answer three questions about the suitability of Deep Learning and Transfer Learning to generate predictions of time series. Time series of M4 and M3 competitions were used for the experiments. The results suggest that deep learning models based on TCN, LSTM, and CNN with transfer learning tend to surpass the performance prediction of other traditional methods. On the other hand, TCN and LSTM, trained directly on the target time series, got similar or better performance than traditional methods for some forecast horizons.

Independent or i.i.d. innovations is an essential assumption in the literature for analyzing a vector time series. However, this assumption is either too restrictive for a real-life time series to satisfy or is hard to verify through a hypothesis test. This paper performs statistical inference on a sparse high-dimensional vector autoregressive time series, allowing its white noise innovations to be dependent, even non-stationary. To achieve this goal, it adopts a post-selection estimator to fit the vector autoregressive model and derives the asymptotic distribution of the post-selection estimator. The innovations in the autoregressive time series are not assumed to be independent, thus making the covariance matrices of the autoregressive coefficient estimators complex and difficult to estimate. Our work develops a bootstrap algorithm to facilitate practitioners in performing statistical inference without having to engage in sophisticated calculations. Simulations and real-life data experiments reveal the validity of the proposed methods and theoretical results. Real-life data is rarely considered to exactly satisfy an autoregressive model with independent or i.i.d. innovations, so our work should better reflect the reality compared to the literature that assumes i.i.d. innovations.

Remotely sensed data are dominated by mixed Land Use and Land Cover (LULC) types. Spectral unmixing is a technique to extract information from mixed pixels into their constituent LULC types and corresponding abundance fractions. Traditionally, solving this task has relied on either classical methods that require prior knowledge of endmembers or machine learning methods that avoid explicit endmembers calculation, also known as blind spectral unmixing (BSU). Most BSU studies based on Deep Learning (DL) focus on one time-step hyperspectral data, yet its acquisition remains quite costly compared with multispectral data. To our knowledge, here we provide the first study on BSU of LULC classes using multispectral time series data with DL models. We further boost the performance of a Long-Short Term Memory (LSTM)-based model by incorporating geographic plus topographic (geo-topographic) and climatic ancillary information. Our experiments show that combining spectral-temporal input data together with geo-topographic and climatic information substantially improves the abundance estimation of LULC classes in mixed pixels. To carry out this study, we built a new labeled dataset of the region of Andalusia (Spain) with monthly multispectral time series of pixels for the year 2013 from MODIS at 460m resolution, for two hierarchical levels of LULC classes, named Andalusia MultiSpectral MultiTemporal Unmixing (Andalusia-MSMTU). This dataset provides, at the pixel level, a multispectral time series plus ancillary information annotated with the abundance of each LULC class inside each pixel. The dataset and code are available to the public.

The generalized Golub-Kahan bidiagonalization has been used to solve saddle-point systems where the leading block is symmetric and positive definite. We extend this iterative method for the case where the symmetry condition no longer holds. We do so by relying on the known connection the algorithm has with the Conjugate Gradient method and following the line of reasoning that adapts the latter into the Full Orthogonalization Method. We propose appropriate stopping criteria based on the residual and an estimate of the energy norm for the error associated with the primal variable. Numerical comparison with GMRES highlights the advantages of our proposed strategy regarding its low memory requirements and the associated implications.