We use tools from random matrix theory to study the multi-spiked tensor model, i.e., a rank-$r$ deformation of a symmetric random Gaussian tensor. In particular, thanks to the nature of local optimization methods used to find the maximum likelihood estimator of this model, we propose to study the phase transition phenomenon for finding critical points of the corresponding optimization problem, i.e., those points defined by the Karush-Kuhn-Tucker (KKT) conditions. Moreover, we characterize the limiting alignments between the estimated signals corresponding to a critical point of the likelihood and the ground truth signals. With the help of these results, we propose a new estimator of the rank-$r$ tensor weights by solving a system of polynomial equations, which is asymptotically unbiased contrary the maximum likelihood estimator.

Abstract--Prenatal ultrasound evaluates fetal growth and detects congenital abnormalities during pregnancy, but the examination of ultrasound images by radiologists requires expertise and sophisticated equipment, which would otherwise fail to improve the rate of identifying specific types of fetal central nervous system (CNS) abnormalities and result in unnecessary patient examinations. We construct a deep learning model to improve the overall accuracy of the diagnosis of fetal cranial anomalies to aid prenatal diagnosis. In our collected multi-center dataset of fetal craniocerebral anomalies covering four typical anomalies of the fetal central nervous system (CNS): anencephaly, encephalocele (including meningocele), holoprosencephaly, and rachischisis, patient-level prediction accuracy reaches 94.5%, with an AUROC value of 99.3%. In the subgroup analyzes, our model is applicable to the entire gestational period, with good identification of fetal anomaly types for any gestational period. Heatmaps superimposed on the ultrasound images not only provide a visual interpretation for the algorithm but also provide an intuitive visual aid to the physician by highlighting key areas that need to be reviewed, helping the physician to quickly identify and validate key areas. Finally, the retrospective reader study demonstrates that by combining the automatic prediction of the DL system with the professional judgment of the radiologist, the diagnostic accuracy and efficiency can be effectively improved and the misdiagnosis rate can be reduced, which has an important clinical application prospect. Optimizing the prenatal ultrasound diagnosis process can significantly reduce Ultrasonography is popular as a non-invasive and radiationfree the workload of the sonographer; therefore, the application of prenatal diagnostic method for its convenience and low artificial intelligence (AI) and deep learning (DL) techniques cost [1]. Antenatal ultrasound is a crucial imaging tool during in ultrasound imaging can significantly speed up the prenatal pregnancy. It not only assesses fetal growth and development examination process while improving the accuracy and consistency and detects congenital anomalies, but also provides important of the diagnosis. Deep learning, a subset of AI, automatically extracts ultrasound, physicians can assess the presence of congenital features from large amounts of data and performs efficient anomalies in the fetus with the help of two-dimensional (2D) pattern recognition and prediction using deep neural network and three-dimensional (3D) imaging, thus helping to significantly models [5].

Building robust, interpretable, and secure artificial intelligence system requires some degree of quantifying and representing uncertainty via a probabilistic perspective, as it allows to mimic human cognitive abilities. However, probabilistic computation presents significant challenges due to its inherent complexity. In this paper, we develop an efficient and interpretable probabilistic computation framework by truncating the probabilistic representation up to its first two moments, i.e., mean and covariance. We instantiate the framework by training a deterministic surrogate of a stochastic network that learns the complex probabilistic representation via combinations of simple activations, encapsulating the non-linearities coupling of the mean and covariance. We show that when the mean is supervised for optimizing the task objective, the unsupervised covariance spontaneously emerging from the non-linear coupling with the mean faithfully captures the uncertainty associated with model predictions. Our research highlights the inherent computability and simplicity of probabilistic computation, enabling its wider application in large-scale settings.

We investigate the geometry of the empirical risk minimization problem for $k$-layer neural networks. We will provide examples showing that for the classical activation functions $\sigma(x)= 1/\bigl(1 + \exp(-x)\bigr)$ and $\sigma(x)=\tanh(x)$, there exists a positive-measured subset of target functions that do not have best approximations by a fixed number of layers of neural networks. In addition, we study in detail the properties of shallow networks, classifying cases when a best $k$-layer neural network approximation always exists or does not exist for the ReLU activation $\sigma=\max(0,x)$. We also determine the dimensions of shallow ReLU-activated networks.

In this paper, we study a polynomial decomposition model that arises in problems of system identification, signal processing and machine learning. We show that this decomposition is a special case of the X-rank decomposition --- a powerful novel concept in algebraic geometry that generalizes the tensor CP decomposition. We prove new results on generic/maximal rank and on identifiability of a particular polynomial decomposition model. In the paper, we try to make results and basic tools accessible for general audience (assuming no knowledge of algebraic geometry or its prerequisites).