Search

On September 16th, 2023, an anomalous 10.88 mHz seismic signal was observed globally, persisting for 9 days. One month later an identical signal appeared, lasting for another week. Several studies have theorized that these signals were produced by seiches which formed after two landslide generated mega-tsunamis in an East-Greenland fjord. This theory is supported by seismic inversions, and analytical and numerical modeling, but no direct observations have been made -- until now. Using data from the new Surface Water Ocean Topography mission, we present the first observations of this phenomenon. By ruling out other oceanographic processes, we validate the seiche theory of previous authors and independently estimate its initial amplitude at 7.9 m using Bayesian machine learning and seismic data. This study demonstrates the value of satellite altimetry for studying extreme events, while also highlighting the need for specialized methods to address the altimetric data's limitations, namely temporal sparsity. These data and approaches will help in understanding future unseen extremes driven by climate change., Comment: 19 pages, 9 figures

Operator learning, the approximation of mappings between infinite-dimensional function spaces using ideas from machine learning, has gained increasing research attention in recent years. Approximate operators, learned from data, hold promise to serve as efficient surrogate models for problems in computational science and engineering, complementing traditional numerical methods. However, despite their empirical success, our understanding of the underpinning mathematical theory is in large part still incomplete. In this paper, we study the approximation of Lipschitz operators in expectation with respect to Gaussian measures. We prove higher Gaussian Sobolev regularity of Lipschitz operators and establish lower and upper bounds on the Hermite polynomial approximation error. We further consider the reconstruction of Lipschitz operators from $m$ arbitrary (adaptive) linear samples. A key finding is the tight characterization of the smallest achievable error for all possible (adaptive) sampling and reconstruction maps in terms of $m$. It is shown that Hermite polynomial approximation is an optimal recovery strategy, but we have the following curse of sample complexity: No method to approximate Lipschitz operators based on finitely many samples can achieve algebraic convergence rates in $m$. On the positive side, we prove that a sufficiently fast spectral decay of the covariance operator of the Gaussian measure guarantees convergence rates which are arbitrarily close to any algebraic rate in the large data limit $m \to \infty$. Finally, we focus on the recovery of Lipschitz operators from finitely many point samples. We consider Christoffel sampling and weighted least-squares approximation, and present an algorithm which provably achieves near-optimal sample complexity., Comment: 56 pages

Least-squares approximation is one of the most important methods for recovering an unknown function from data. While in many applications the data is fixed, in many others there is substantial freedom to choose where to sample. In this paper, we review recent progress on optimal sampling for (weighted) least-squares approximation in arbitrary linear spaces. We introduce the Christoffel function as a key quantity in the analysis of (weighted) least-squares approximation from random samples, then show how it can be used to construct sampling strategies that possess near-optimal sample complexity: namely, the number of samples scales log-linearly in $n$, the dimension of the approximation space. We discuss a series of variations, extensions and further topics, and throughout highlight connections to approximation theory, machine learning, information-based complexity and numerical linear algebra. Finally, motivated by various contemporary applications, we consider a generalization of the classical setting where the samples need not be pointwise samples of a scalar-valued function, and the approximation space need not be linear. We show that even in this significantly more general setting suitable generalizations of the Christoffel function still determine the sample complexity. This provides a unified procedure for designing improved sampling strategies for general recovery problems. This article is largely self-contained, and intended to be accessible to nonspecialists.

We introduce the Consistent Reasoning Paradox (CRP). Consistent reasoning, which lies at the core of human intelligence, is the ability to handle tasks that are equivalent, yet described by different sentences ('Tell me the time!' and 'What is the time?'). The CRP asserts that consistent reasoning implies fallibility -- in particular, human-like intelligence in AI necessarily comes with human-like fallibility. Specifically, it states that there are problems, e.g. in basic arithmetic, where any AI that always answers and strives to mimic human intelligence by reasoning consistently will hallucinate (produce wrong, yet plausible answers) infinitely often. The paradox is that there exists a non-consistently reasoning AI (which therefore cannot be on the level of human intelligence) that will be correct on the same set of problems. The CRP also shows that detecting these hallucinations, even in a probabilistic sense, is strictly harder than solving the original problems, and that there are problems that an AI may answer correctly, but it cannot provide a correct logical explanation for how it arrived at the answer. Therefore, the CRP implies that any trustworthy AI (i.e., an AI that never answers incorrectly) that also reasons consistently must be able to say 'I don't know'. Moreover, this can only be done by implicitly computing a new concept that we introduce, termed the 'I don't know' function -- something currently lacking in modern AI. In view of these insights, the CRP also provides a glimpse into the behaviour of Artificial General Intelligence (AGI). An AGI cannot be 'almost sure', nor can it always explain itself, and therefore to be trustworthy it must be able to say 'I don't know'., Comment: 12 pages and 50 pages of supplementary material, 7 figures

Operator learning problems arise in many key areas of scientific computing where Partial Differential Equations (PDEs) are used to model physical systems. In such scenarios, the operators map between Banach or Hilbert spaces. In this work, we tackle the problem of learning operators between Banach spaces, in contrast to the vast majority of past works considering only Hilbert spaces. We focus on learning holomorphic operators - an important class of problems with many applications. We combine arbitrary approximate encoders and decoders with standard feedforward Deep Neural Network (DNN) architectures - specifically, those with constant width exceeding the depth - under standard $\ell^2$-loss minimization. We first identify a family of DNNs such that the resulting Deep Learning (DL) procedure achieves optimal generalization bounds for such operators. For standard fully-connected architectures, we then show that there are uncountably many minimizers of the training problem that yield equivalent optimal performance. The DNN architectures we consider are `problem agnostic', with width and depth only depending on the amount of training data $m$ and not on regularity assumptions of the target operator. Next, we show that DL is optimal for this problem: no recovery procedure can surpass these generalization bounds up to log terms. Finally, we present numerical results demonstrating the practical performance on challenging problems including the parametric diffusion, Navier-Stokes-Brinkman and Boussinesq PDEs.

Learning approximations to smooth target functions of many variables from finite sets of pointwise samples is an important task in scientific computing and its many applications in computational science and engineering. Despite well over half a century of research on high-dimensional approximation, this remains a challenging problem. Yet, significant advances have been made in the last decade towards efficient methods for doing this, commencing with so-called sparse polynomial approximation methods and continuing most recently with methods based on Deep Neural Networks (DNNs). In tandem, there have been substantial advances in the relevant approximation theory and analysis of these techniques. In this work, we survey this recent progress. We describe the contemporary motivations for this problem, which stem from parametric models and computational uncertainty quantification; the relevant function classes, namely, classes of infinite-dimensional, Banach-valued, holomorphic functions; fundamental limits of learnability from finite data for these classes; and finally, sparse polynomial and DNN methods for efficiently learning such functions from finite data. For the latter, there is currently a significant gap between the approximation theory of DNNs and the practical performance of deep learning. Aiming to narrow this gap, we develop the topic of practical existence theory, which asserts the existence of dimension-independent DNN architectures and training strategies that achieve provably near-optimal generalization errors in terms of the amount of training data.

Heralded photons from a silicon source are temporally multiplexed utilizing thin film lithium niobate photonics. The time-multiplexed source, operating at a rate of R = 62.2 MHz, enhances single photon probability by 3.25 $\pm$ 0.05.

We present direct numerical simulations of the splashing process between two cylindrical liquid rims. This belongs to a class of impact and collision problems with a wide range of applications in science and engineering, and motivated here by splashing of breaking ocean waves. Interfacial perturbations with a truncated white noise frequency profile are introduced to the rims before their collision, whose subsequent morphological development is simulated by solving the two-phase incompressible Navier-Stokes equation with the adaptive mesh refinement (AMR) technique, within the Basilisk software environment. We first derive analytical solutions predicting the unsteady interfacial and velocity profiles of the expanding sheet forming between the two rims, and develop scaling laws for the evolution of the lamella rim under capillary deceleration. We then analyse the formation and growth of transverse ligaments ejected from the lamella rims, which we find to originate from the initial corrugated geometry of the perturbed rim surface. Novel scaling models are proposed for predicting the decay of the ligament number density due to the ongoing ligament merging phenomenon, and found to agree well with the numerical results presented here. The role of the mechanism in breaking waves is discussed further and necessary next steps in the problem are identified.

Infinite-dimensional, holomorphic functions have been studied in detail over the last several decades, due to their relevance to parametric differential equations and computational uncertainty quantification. The approximation of such functions from finitely many samples is of particular interest, due to the practical importance of constructing surrogate models to complex mathematical models of physical processes. In a previous work, [5] we studied the approximation of so-called Banach-valued, $(\boldsymbol{b},\varepsilon)$-holomorphic functions on the infinite-dimensional hypercube $[-1,1]^{\mathbb{N}}$ from $m$ (potentially adaptive) samples. In particular, we derived lower bounds for the adaptive $m$-widths for classes of such functions, which showed that certain algebraic rates of the form $m^{1/2-1/p}$ are the best possible regardless of the sampling-recovery pair. In this work, we continue this investigation by focusing on the practical case where the samples are pointwise evaluations drawn identically and independently from a probability measure. Specifically, for Hilbert-valued $(\boldsymbol{b},\varepsilon)$-holomorphic functions, we show that the same rates can be achieved (up to a small polylogarithmic or algebraic factor) for essentially arbitrary tensor-product Jacobi (ultraspherical) measures. Our reconstruction maps are based on least squares and compressed sensing procedures using the corresponding orthonormal Jacobi polynomials. In doing so, we strengthen and generalize past work that has derived weaker nonuniform guarantees for the uniform and Chebyshev measures (and corresponding polynomials) only. We also extend various best $s$-term polynomial approximation error bounds to arbitrary Jacobi polynomial expansions. Overall, we demonstrate that i.i.d.\ pointwise samples are near-optimal for the recovery of infinite-dimensional, holomorphic functions.

Computer vision models have known performance disparities across attributes such as gender and skin tone. This means during tasks such as classification and detection, model performance differs for certain classes based on the demographics of the people in the image. These disparities have been shown to exist, but until now there has not been a unified approach to measure these differences for common use-cases of computer vision models. We present a new benchmark named FACET (FAirness in Computer Vision EvaluaTion), a large, publicly available evaluation set of 32k images for some of the most common vision tasks - image classification, object detection and segmentation. For every image in FACET, we hired expert reviewers to manually annotate person-related attributes such as perceived skin tone and hair type, manually draw bounding boxes and label fine-grained person-related classes such as disk jockey or guitarist. In addition, we use FACET to benchmark state-of-the-art vision models and present a deeper understanding of potential performance disparities and challenges across sensitive demographic attributes. With the exhaustive annotations collected, we probe models using single demographics attributes as well as multiple attributes using an intersectional approach (e.g. hair color and perceived skin tone). Our results show that classification, detection, segmentation, and visual grounding models exhibit performance disparities across demographic attributes and intersections of attributes. These harms suggest that not all people represented in datasets receive fair and equitable treatment in these vision tasks. We hope current and future results using our benchmark will contribute to fairer, more robust vision models. FACET is available publicly at https://facet.metademolab.com/

We introduce a general framework for active learning in regression problems. Our framework extends the standard setup by allowing for general types of data, rather than merely pointwise samples of the target function. This generalization covers many cases of practical interest, such as data acquired in transform domains (e.g., Fourier data), vector-valued data (e.g., gradient-augmented data), data acquired along continuous curves, and, multimodal data (i.e., combinations of different types of measurements). Our framework considers random sampling according to a finite number of sampling measures and arbitrary nonlinear approximation spaces (model classes). We introduce the concept of generalized Christoffel functions and show how these can be used to optimize the sampling measures. We prove that this leads to near-optimal sample complexity in various important cases. This paper focuses on applications in scientific computing, where active learning is often desirable, since it is usually expensive to generate data. We demonstrate the efficacy of our framework for gradient-augmented learning with polynomials, Magnetic Resonance Imaging (MRI) using generative models and adaptive sampling for solving PDEs using Physics-Informed Neural Networks (PINNs).

Over the last decade, approximating functions in infinite dimensions from samples has gained increasing attention in computational science and engineering, especially in computational uncertainty quantification. This is primarily due to the relevance of functions that are solutions to parametric differential equations in various fields, e.g. chemistry, economics, engineering, and physics. While acquiring accurate and reliable approximations of such functions is inherently difficult, current benchmark methods exploit the fact that such functions often belong to certain classes of holomorphic functions to get algebraic convergence rates in infinite dimensions with respect to the number of (potentially adaptive) samples $m$. Our work focuses on providing theoretical approximation guarantees for the class of $(\boldsymbol{b},\varepsilon)$-holomorphic functions, demonstrating that these algebraic rates are the best possible for Banach-valued functions in infinite dimensions. We establish lower bounds using a reduction to a discrete problem in combination with the theory of $m$-widths, Gelfand widths and Kolmogorov widths. We study two cases, known and unknown anisotropy, in which the relative importance of the variables is known and unknown, respectively. A key conclusion of our paper is that in the latter setting, approximation from finite samples is impossible without some inherent ordering of the variables, even if the samples are chosen adaptively. Finally, in both cases, we demonstrate near-optimal, non-adaptive (random) sampling and recovery strategies which achieve close to same rates as the lower bounds.

This paper revisits the standard pretrain-then-finetune paradigm used in computer vision for visual recognition tasks. Typically, state-of-the-art foundation models are pretrained using large scale (weakly) supervised datasets with billions of images. We introduce an additional pre-pretraining stage that is simple and uses the self-supervised MAE technique to initialize the model. While MAE has only been shown to scale with the size of models, we find that it scales with the size of the training dataset as well. Thus, our MAE-based pre-pretraining scales with both model and data size making it applicable for training foundation models. Pre-pretraining consistently improves both the model convergence and the downstream transfer performance across a range of model scales (millions to billions of parameters), and dataset sizes (millions to billions of images). We measure the effectiveness of pre-pretraining on 10 different visual recognition tasks spanning image classification, video recognition, object detection, low-shot classification and zero-shot recognition. Our largest model achieves new state-of-the-art results on iNaturalist-18 (91.7%), ImageNet-ReaL (91.1%), 1-shot ImageNet-1k (63.6%), and zero-shot transfer on Food-101 (96.2%). Our study reveals that model initialization plays a significant role, even for web-scale pretraining with billions of images, and our models are available publicly., Comment: ICCV 2023. Models available at https://github.com/facebookresearch/maws/

We explore the extent to which zero-shot vision-language models exhibit gender bias for different vision tasks. Vision models traditionally required task-specific labels for representing concepts, as well as finetuning; zero-shot models like CLIP instead perform tasks with an open-vocabulary, meaning they do not need a fixed set of labels, by using text embeddings to represent concepts. With these capabilities in mind, we ask: Do vision-language models exhibit gender bias when performing zero-shot image classification, object detection and semantic segmentation? We evaluate different vision-language models with multiple datasets across a set of concepts and find (i) all models evaluated show distinct performance differences based on the perceived gender of the person co-occurring with a given concept in the image and that aggregating analyses over all concepts can mask these concerns; (ii) model calibration (i.e. the relationship between accuracy and confidence) also differs distinctly by perceived gender, even when evaluating on similar representations of concepts; and (iii) these observed disparities align with existing gender biases in word embeddings from language models. These findings suggest that, while language greatly expands the capability of vision tasks, it can also contribute to social biases in zero-shot vision settings. Furthermore, biases can further propagate when foundational models like CLIP are used by other models to enable zero-shot capabilities.

We experimentally demonstrate a room-temperature, voltage controlled, short-term quantum photonics memory on a lithium niobate chip. Our chip is capable of resolving 100 ps time steps with 0.74 dB loss per round-trip.

Current dataset collection methods typically scrape large amounts of data from the web. While this technique is extremely scalable, data collected in this way tends to reinforce stereotypical biases, can contain personally identifiable information, and typically originates from Europe and North America. In this work, we rethink the dataset collection paradigm and introduce GeoDE, a geographically diverse dataset with 61,940 images from 40 classes and 6 world regions, and no personally identifiable information, collected through crowd-sourcing. We analyse GeoDE to understand differences in images collected in this manner compared to web-scraping. Despite the smaller size of this dataset, we demonstrate its use as both an evaluation and training dataset, highlight shortcomings in current models, as well as show improved performances when even small amounts of GeoDE (1000 - 2000 images per region) are added to a training dataset. We release the full dataset and code at https://geodiverse-data-collection.cs.princeton.edu/

Sharpness is an almost generic assumption in continuous optimization that bounds the distance from minima by objective function suboptimality. It facilitates the acceleration of first-order methods through restarts. However, sharpness involves problem-specific constants that are typically unknown, and restart schemes typically reduce convergence rates. Moreover, these schemes are challenging to apply in the presence of noise or with approximate model classes (e.g., in compressive imaging or learning problems), and they generally assume that the first-order method used produces feasible iterates. We consider the assumption of approximate sharpness, a generalization of sharpness that incorporates an unknown constant perturbation to the objective function error. This constant offers greater robustness (e.g., with respect to noise or relaxation of model classes) for finding approximate minimizers. By employing a new type of search over the unknown constants, we design a restart scheme that applies to general first-order methods and does not require the first-order method to produce feasible iterates. Our scheme maintains the same convergence rate as when the constants are known. The convergence rates we achieve for various first-order methods match the optimal rates or improve on previously established rates for a wide range of problems. We showcase our restart scheme in several examples and highlight potential future applications and developments of our framework and theory., Comment: Version accepted in Foundations of Computational Mathematics

We present novel numerical simulations investigating the bag breakup of liquid droplets. We first examine the viscous effect on drop deformation before the onset of bag breakup, comparing with theory and experiment. Next, a bag film forms at late time and is susceptible to spurious mesh-induced breakup in numerical simulations, which has prevented previous studies from reaching grid convergence of fragment statistics. We therefore adopt the manifold death (MD) algorithm which artificially perforates thin films once they reach a prescribed critical thickness independent of the grid size, controlled by a numerical parameter $L_{\rm sig}$. We show grid convergence of fragment statistics when utilising the MD algorithm, and analyse the fragment behaviour and bag film disintegration mechanisms including ligament breakup, node detachment and rim destabilisation. Our choice of the critical thickness parameter $L_{\rm sig}$ is limited by numerical constraints and thus has not been matched to experiment or theory; consequently, the current simulations yield critical bag film perforation thicknesses larger than experimentally observed. The influence of the MD algorithm configuration on the bag breakup phenomena and statistics will be investigated in future work. We also study the effects of moderate liquid Ohnesorge number ($0.005 \leq Oh \leq 0.05$) on the bag breakup process and fragment statistics, where a non-monotonic dependency of the average diameter of bag film fragments on $Oh$ is found. These results highlight the utility of the MD algorithm in multiphase simulations involving topological changes, and pave the way for physics-based numerical investigations into spume generation at the air-sea interface., Comment: Fourth Submission

The past decade has seen increasing interest in applying Deep Learning (DL) to Computational Science and Engineering (CSE). Driven by impressive results in applications such as computer vision, Uncertainty Quantification (UQ), genetics, simulations and image processing, DL is increasingly supplanting classical algorithms, and seems poised to revolutionize scientific computing. However, DL is not yet well-understood from the standpoint of numerical analysis. Little is known about the efficiency and reliability of DL from the perspectives of stability, robustness, accuracy, and sample complexity. In particular, approximating solutions to parametric PDEs is an objective of UQ for CSE. Training data for such problems is often scarce and corrupted by errors. Moreover, the target function is a possibly infinite-dimensional smooth function taking values in the PDE solution space, generally an infinite-dimensional Banach space. This paper provides arguments for Deep Neural Network (DNN) approximation of such functions, with both known and unknown parametric dependence, that overcome the curse of dimensionality. We establish practical existence theorems that describe classes of DNNs with dimension-independent architecture size and training procedures based on minimizing the (regularized) $\ell^2$-loss which achieve near-optimal algebraic rates of convergence. These results involve key extensions of compressed sensing for Banach-valued recovery and polynomial emulation with DNNs. When approximating solutions of parametric PDEs, our results account for all sources of error, i.e., sampling, optimization, approximation and physical discretization, and allow for training high-fidelity DNN approximations from coarse-grained sample data. Our theoretical results fall into the category of non-intrusive methods, providing a theoretical alternative to classical methods for high-dimensional approximation., Comment: 55 pages

The computation of global radial basis function (RBF) approximations requires the solution of a linear system which, depending on the choice of RBF parameters, may be ill-conditioned. We study the stability and accuracy of approximation methods using the Gaussian RBF in all scaling regimes of the associated shape parameter. The approximation is based on discrete least squares with function samples on a bounded domain, using RBF centers both inside and outside the domain. This results in a rectangular linear system. We show for one-dimensional approximations that linear scaling of the shape parameter with the degrees of freedom is optimal, resulting in constant overlap between neighbouring RBF's regardless of their number, and we propose an explicit suitable choice of the proportionality constant. We show numerically that highly accurate approximations to smooth functions can also be obtained on bounded domains in several dimensions, using a linear scaling with the degrees of freedom per dimension. We extend the least squares approach to a collocation-based method for the solution of elliptic boundary value problems and illustrate that the combination of centers outside the domain, oversampling and optimal scaling can result in accuracy close to machine precision in spite of having to solve very ill-conditioned linear systems.

Grid-interactive building control is a challenging and important problem for reducing carbon emissions, increasing energy efficiency, and supporting the electric power grid. Currently researchers and practitioners are confronted with a choice of control strategies ranging from model-free (purely data-driven) to model-based (directly incorporating physical knowledge) to hybrid methods that combine data and models. In this work, we identify state-of-the-art methods that span this methodological spectrum and evaluate their performance for multi-zone building HVAC control in the context of three demand response programs. We demonstrate, in this context, that hybrid methods offer many benefits over both purely model-free and model-based methods as long as certain requirements are met. In particular, hybrid controllers are relatively sample efficient, fast online, and high accuracy so long as the test case falls within the distribution of training data. Like all data-driven methods, hybrid controllers are still subject to generalization errors when applied to out-of-sample scenarios. Key takeaways for control strategies are summarized and the developed software framework is open-sourced.

The problem of approximating smooth, multivariate functions from sample points arises in many applications in scientific computing, e.g., in computational Uncertainty Quantification (UQ) for science and engineering. In these applications, the target function may represent a desired quantity of interest of a parameterized Partial Differential Equation (PDE). Due to the large cost of solving such problems, where each sample is computed by solving a PDE, sample efficiency is a key concerning these applications. Recently, there has been increasing focus on the use of Deep Neural Networks (DNN) and Deep Learning (DL) for learning such functions from data. In this work, we propose an adaptive sampling strategy, CAS4DL (Christoffel Adaptive Sampling for Deep Learning) to increase the sample efficiency of DL for multivariate function approximation. Our novel approach is based on interpreting the second to last layer of a DNN as a dictionary of functions defined by the nodes on that layer. With this viewpoint, we then define an adaptive sampling strategy motivated by adaptive sampling schemes recently proposed for linear approximation schemes, wherein samples are drawn randomly with respect to the Christoffel function of the subspace spanned by this dictionary. We present numerical experiments comparing CAS4DL with standard Monte Carlo (MC) sampling. Our results demonstrate that CAS4DL often yields substantial savings in the number of samples required to achieve a given accuracy, particularly in the case of smooth activation functions, and it shows a better stability in comparison to MC. These results therefore are a promising step towards fully adapting DL towards scientific computing applications., Comment: 34 pages, 9 figures

This paper concerns the approximation of smooth, high-dimensional functions from limited samples using polynomials. This task lies at the heart of many applications in computational science and engineering - notably, some of those arising from parametric modelling and computational uncertainty quantification. It is common to use Monte Carlo sampling in such applications, so as not to succumb to the curse of dimensionality. However, it is well known that such a strategy is theoretically suboptimal. Specifically, there are many polynomial spaces of dimension $n$ for which the sample complexity scales log-quadratically, i.e., like $c \cdot n^2 \cdot \log(n)$ as $n \rightarrow \infty$. This well-documented phenomenon has led to a concerted effort over the last decade to design improved, and moreover, near-optimal strategies, whose sample complexities scale log-linearly, or even linearly in $n$. In this work we demonstrate that Monte Carlo is actually a perfectly good strategy in high dimensions, despite its apparent suboptimality. We first document this phenomenon empirically via a systematic set of numerical experiments. Next, we present a theoretical analysis that rigorously justifies this fact in the case of holomorphic functions of infinitely-many variables. We show that there is a least-squares approximation based on $m$ Monte Carlo samples whose error decays algebraically fast in $m/\log(m)$, with a rate that is the same as that of the best $n$-term polynomial approximation. This result is non-constructive, since it assumes knowledge of a suitable polynomial subspace in which to perform the approximation. We next present a compressed sensing-based scheme that achieves the same rate, except for a larger polylogarithmic factor. This scheme is practical, and numerically it performs as well as or better than well-known adaptive least-squares schemes.

Efficient generation of single photons is one of the key challenges of building photonic quantum technology, such as quantum computers and long-distance quantum networks. Photon source multiplexing -- where successful pair generation is heralded by the detection of one of the photons, and its partner is routed to a single mode output -- has long been known to offer a concrete solution, with output probability tending toward unity as loss is reduced. Here, we present a temporally multiplexed integrated single photon source based on a silicon waveguide and a low-loss fibre switch and loop architecture, which achieves enhancement of the single photon output probability of $4.5 \pm 0.5$, while retaining $g^{(2)}(0) = 0.01$.

Quantum technology is poised to enable a step change in human capability for computing, communications and sensing. Photons are indispensable as carriers of quantum information - they travel at the fastest possible speed and readily protected from decoherence. However, the system requires thousands of near-transparent components with ultra-low-latency control. For quantum technology to be implemented, a new paradigm photonic system is required: one with in-built coherence, stability, the ability to define arbitrary circuits, and a path to manufacturability. Silicon photonics has unparalleled density and component performance, which, with CMOS compatible fabrication, place it in a strong position for a scalable quantum photonics platform. This paper is a progress report on silicon quantum photonics, focused on developments in the past five years. We provide an introduction on silicon quantum photonic component and the challenges in the field, summarise the current state-of-the-art and identify outstanding technical challenges, as well as promising avenues of future research. We also resolve a conflict in the definition of Hong-Ou-Mandel interference visibility in integrated quantum photonic experiments, needed for fair comparison of photon quality across different platforms. Our aim is the development of scalability on the platform, to which end we point the way to ever-closer integration, toward silicon quantum photonic systems-on-a-chip.