Iterative decoder failures of quantum low density parity check (QLDPC) codes are attributed to substructures in the code's graph, known as trapping sets, as well as degenerate errors that can arise in quantum codes. Failure inducing sets are subsets of codeword coordinates that, when initially in error, lead to decoding failure in a trapping set. In this paper we examine the failure inducing sets of QLDPC codes under syndrome-based iterative decoding, and their connecting to absorbing sets in classical LDPC codes.
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Smoothing splines are twice differentiable by construction, so they cannot capture potential discontinuities in the underlying signal. In this work, we consider a special case of the weak rod model of Blake and Zisserman (1987) that allows for discontinuities penalizing their number by a linear term. The corresponding estimates are cubic smoothing splines with discontinuities (CSSD) which serve as representations of piecewise smooth signals and facilitate exploratory data analysis. However, computing the estimates requires solving a non-convex optimization problem. So far, efficient and exact solvers exist only for a discrete approximation based on equidistantly sampled data. In this work, we propose an efficient solver for the continuous minimization problem with non-equidistantly sampled data. Its worst case complexity is quadratic in the number of data points, and if the number of detected discontinuities scales linearly with the signal length, we observe linear growth in runtime. This efficient algorithm allows to use cross validation for automatic selection of the hyperparameters within a reasonable time frame on standard hardware. We provide a reference implementation and supplementary material. We demonstrate the applicability of the approach for the aforementioned tasks using both simulated and real data.
Three-dimensional facial stereophotogrammetry provides a detailed representation of craniofacial soft tissue without the use of ionizing radiation. While manual annotation of landmarks serves as the current gold standard for cephalometric analysis, it is a time-consuming process and is prone to human error. The aim in this study was to develop and evaluate an automated cephalometric annotation method using a deep learning-based approach. Ten landmarks were manually annotated on 2897 3D facial photographs by a single observer. The automated landmarking workflow involved two successive DiffusionNet models and additional algorithms for facial segmentation. The dataset was randomly divided into a training and test dataset. The training dataset was used to train the deep learning networks, whereas the test dataset was used to evaluate the performance of the automated workflow. The precision of the workflow was evaluated by calculating the Euclidean distances between the automated and manual landmarks and compared to the intra-observer and inter-observer variability of manual annotation and the semi-automated landmarking method. The workflow was successful in 98.6% of all test cases. The deep learning-based landmarking method achieved precise and consistent landmark annotation. The mean precision of 1.69 (+/-1.15) mm was comparable to the inter-observer variability (1.31 +/-0.91 mm) of manual annotation. The Euclidean distance between the automated and manual landmarks was within 2 mm in 69%. Automated landmark annotation on 3D photographs was achieved with the DiffusionNet-based approach. The proposed method allows quantitative analysis of large datasets and may be used in diagnosis, follow-up, and virtual surgical planning.
Many modern discontinuous Galerkin (DG) methods for conservation laws make use of summation by parts operators and flux differencing to achieve kinetic energy preservation or entropy stability. While these techniques increase the robustness of DG methods significantly, they are also computationally more demanding than standard weak form nodal DG methods. We present several implementation techniques to improve the efficiency of flux differencing DG methods that use tensor product quadrilateral or hexahedral elements, in 2D or 3D respectively. Focus is mostly given to CPUs and DG methods for the compressible Euler equations, although these techniques are generally also useful for other physical systems including the compressible Navier-Stokes and magnetohydrodynamics equations. We present results using two open source codes, Trixi.jl written in Julia and FLUXO written in Fortran, to demonstrate that our proposed implementation techniques are applicable to different code bases and programming languages.
Ordinary differential equations (ODEs) are widely used to model complex dynamics that arises in biology, chemistry, engineering, finance, physics, etc. Calibration of a complicated ODE system using noisy data is generally very difficult. In this work, we propose a two-stage nonparametric approach to address this problem. We first extract the de-noised data and their higher order derivatives using boundary kernel method, and then feed them into a sparsely connected deep neural network with ReLU activation function. Our method is able to recover the ODE system without being subject to the curse of dimensionality and complicated ODE structure. When the ODE possesses a general modular structure, with each modular component involving only a few input variables, and the network architecture is properly chosen, our method is proven to be consistent. Theoretical properties are corroborated by an extensive simulation study that demonstrates the validity and effectiveness of the proposed method. Finally, we use our method to simultaneously characterize the growth rate of Covid-19 infection cases from 50 states of the USA.
We present a robust deep incremental learning framework for regression tasks on financial temporal tabular datasets which is built upon the incremental use of commonly available tabular and time series prediction models to adapt to distributional shifts typical of financial datasets. The framework uses a simple basic building block (decision trees) to build self-similar models of any required complexity to deliver robust performance under adverse situations such as regime changes, fat-tailed distributions, and low signal-to-noise ratios. As a detailed study, we demonstrate our scheme using XGBoost models trained on the Numerai dataset and show that a two layer deep ensemble of XGBoost models over different model snapshots delivers high quality predictions under different market regimes. We also show that the performance of XGBoost models with different number of boosting rounds in three scenarios (small, standard and large) is monotonically increasing with respect to model size and converges towards the generalisation upper bound. We also evaluate the robustness of the model under variability of different hyperparameters, such as model complexity and data sampling settings. Our model has low hardware requirements as no specialised neural architectures are used and each base model can be independently trained in parallel.
Stochastic memoization is a higher-order construct of probabilistic programming languages that is key in Bayesian nonparametrics, a modular approach that allows us to extend models beyond their parametric limitations and compose them in an elegant and principled manner. Stochastic memoization is simple and useful in practice, but semantically elusive, particularly regarding dataflow transformations. As the naive implementation resorts to the state monad, which is not commutative, it is not clear if stochastic memoization preserves the dataflow property -- i.e., whether we can reorder the lines of a program without changing its semantics, provided the dataflow graph is preserved. In this paper, we give an operational and categorical semantics to stochastic memoization and name generation in the context of a minimal probabilistic programming language, for a restricted class of functions. Our contribution is a first model of stochastic memoization of constant Bernoulli functions with a non-enumerable type, which validates data flow transformations, bridging the gap between traditional probability theory and higher-order probability models. Our model uses a presheaf category and a novel probability monad on it.
Transformer-based models, exemplified by GPT-3, ChatGPT, and GPT-4, have recently garnered considerable attention in both academia and industry due to their promising performance in general language tasks. Nevertheless, these models typically involve computationally encoding processes, and in some cases, decoding processes as well, both of which are fundamentally large-scale matrix multiplication. These operations bring the inevitable challenges of massive computation resources and huge memory footprint, usually requiring at least 10^23 FLOPs and hundreds of gigabytes, respectively. A common method to address this issue is to reduce the computational and memory requirements by applying layerwise quantization to the transformer, replacing the usual fp32 data type with a low-bit equivalent. Unfortunately, this method often leads to decreased model accuracy and necessitates time-consuming retraining. Such retraining not only requires fine-tuning skills but also substantial computational resources, posing challenges for users. To specifically tackle these issues, we propose BCT, a framework of blockwise compression for transformers without retraining, aiming to facilitate model deployment. Unlike layerwise compression methods, BCT achieves finer compression of the entire transformer by operating blockwise. This method mitigates data distribution deviation caused by quantization, eliminating the requirement for retraining. BCT effectively compresses all components of the model, including but not limited to the embedding, matrix multiplication, GELU, Softmax, layer normalization, and intermediate results. In a case study, an efficient model is compressed by BCT achieving up to 7.988x compression. Subsequently, we also evaluate it on several General Language Understanding Evaluation (GLUE) datasets.
This paper develops and analyzes an accelerated proximal descent method for finding stationary points of nonconvex composite optimization problems. The objective function is of the form $f+h$ where $h$ is a proper closed convex function, $f$ is a differentiable function on the domain of $h$, and $\nabla f$ is Lipschitz continuous on the domain of $h$. The main advantage of this method is that it is "parameter-free" in the sense that it does not require knowledge of the Lipschitz constant of $\nabla f$ or of any global topological properties of $f$. It is shown that the proposed method can obtain an $\varepsilon$-approximate stationary point with iteration complexity bounds that are optimal, up to logarithmic terms over $\varepsilon$, in both the convex and nonconvex settings. Some discussion is also given about how the proposed method can be leveraged in other existing optimization frameworks, such as min-max smoothing and penalty frameworks for constrained programming, to create more specialized parameter-free methods. Finally, numerical experiments are presented to support the practical viability of the method.
Pretrial risk assessment tools are used in jurisdictions across the country to assess the likelihood of "pretrial failure," the event where defendants either fail to appear for court or reoffend. Judicial officers, in turn, use these assessments to determine whether to release or detain defendants during trial. While algorithmic risk assessment tools were designed to predict pretrial failure with greater accuracy relative to judges, there is still concern that both risk assessment recommendations and pretrial decisions are biased against minority groups. In this paper, we develop methods to investigate the association between risk factors and pretrial failure, while simultaneously estimating misclassification rates of pretrial risk assessments and of judicial decisions as a function of defendant race. This approach adds to a growing literature that makes use of outcome misclassification methods to answer questions about fairness in pretrial decision-making. We give a detailed simulation study for our proposed methodology and apply these methods to data from the Virginia Department of Criminal Justice Services. We estimate that the VPRAI algorithm has near-perfect specificity, but its sensitivity differs by defendant race. Judicial decisions also display evidence of bias; we estimate wrongful detention rates of 39.7% and 51.4% among white and Black defendants, respectively.
Optimal transport and Wasserstein distances are flourishing in many scientific fields as a means for comparing and connecting random structures. Here we pioneer the use of an optimal transport distance between L\'{e}vy measures to solve a statistical problem. Dependent Bayesian nonparametric models provide flexible inference on distinct, yet related, groups of observations. Each component of a vector of random measures models a group of exchangeable observations, while their dependence regulates the borrowing of information across groups. We derive the first statistical index of dependence in $[0,1]$ for (completely) random measures that accounts for their whole infinite-dimensional distribution, which is assumed to be equal across different groups. This is accomplished by using the geometric properties of the Wasserstein distance to solve a max-min problem at the level of the underlying L\'{e}vy measures. The Wasserstein index of dependence sheds light on the models' deep structure and has desirable properties: (i) it is $0$ if and only if the random measures are independent; (ii) it is $1$ if and only if the random measures are completely dependent; (iii) it simultaneously quantifies the dependence of $d \ge 2$ random measures, avoiding the need for pairwise comparisons; (iv) it can be evaluated numerically. Moreover, the index allows for informed prior specifications and fair model comparisons for Bayesian nonparametric models.
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