The measure captures errors due to absent predicted requests as well as unpredicted actual requests; hence, predicted and actual inputs can be of arbitrary size.

There are a number of approximation algorithms for NP-hard versions of low rank approximation, such as finding a rank-$k$ matrix $B$ minimizing the sum of absolute values of differences to a given $n$-by-$n$ matrix $A$, $\min_{\textrm{rank-}k~B}\|A-B\|_1$, or more generally finding a rank-$k$ matrix $B$ which minimizes the sum of $p$-th powers of absolute values of differences, $\min_{\textrm{rank-}k~B}\|A-B\|_p^p$. Many of these algorithms are linear time columns subset selection algorithms, returning a subset of $\poly(k \log n)$ columns whose cost is no more than a $\poly(k)$ factor larger than the cost of the best rank-$k$ matrix.

We also propose a new algorithm that learns to estimate metric depth using annotations of relative depth. Compared to the state of the art, our algorithm is simpler and performs better.

We study the secure stochastic convex optimization problem.

First provide a summary of the paper, and then address the following criteria: Quality, clarity, originality and significance. The authors present a new method for estimating the depth map of a scene using a single image. They use two CNNs: the first outputs a coarse depth prediction based on the entire image. This coarse depth prediction is used as the input, together with the original image, to the second CNN which refines the depth prediction locally. They also present and apply a scale-invariant error measure for the depth prediction.

We introduce a novel measure for quantifying the error in input predictions.

There are a number of approximation algorithms for NP-hard versions of low rank approximation, such as finding a rank- k matrix B minimizing the sum of absolute values of differences to a given n -by- n matrix A, \min_{\textrm{rank-}k B}\ A-B\ _1, or more generally finding a rank- k matrix B which minimizes the sum of p -th powers of absolute values of differences, \min_{\textrm{rank-}k B}\ A-B\ _p p . Many of these algorithms are linear time columns subset selection algorithms, returning a subset of \poly(k \log n) columns whose cost is no more than a \poly(k) factor larger than the cost of the best rank- k matrix. The above error measures are special cases of the following general entrywise low rank approximation problem: given an arbitrary function g:\mathbb{R} \rightarrow \mathbb{R}_{\geq 0}, find a rank- k matrix B which minimizes \ A-B\ _g \sum_{i,j}g(A_{i,j}-B_{i,j}) . A natural question is which functions g admit efficient approximation algorithms? Indeed, this is a central question of recent work studying generalized low rank models.

Purpose: Simulation-based digital twins represent an effort to provide high-accuracy real-time insights into operational physical processes. However, the computation time of many multi-physical simulation models is far from real-time. It might even exceed sensible time frames to produce sufficient data for training data-driven reduced-order models. This study presents TwinLab, a framework for data-efficient, yet accurate training of neural-ODE type reduced-order models with only two data sets. Design/methodology/approach: Correlations between test errors of reduced-order models and distinct features of corresponding training data are investigated. Having found the single best data sets for training, a second data set is sought with the help of similarity and error measures to enrich the training process effectively. Findings: Adding a suitable second training data set in the training process reduces the test error by up to 49% compared to the best base reduced-order model trained only with one data set. Such a second training data set should at least yield a good reduced-order model on its own and exhibit higher levels of dissimilarity to the base training data set regarding the respective excitation signal. Moreover, the base reduced-order model should have elevated test errors on the second data set. The relative error of the time series ranges from 0.18% to 0.49%. Prediction speed-ups of up to a factor of 36,000 are observed. Originality: The proposed computational framework facilitates the automated, data-efficient extraction of non-intrusive reduced-order models for digital twins from existing simulation models, independent of the simulation software.