3D freeform surfaces from planar sketches using neural networks

A novel intelligent approach into 3D freeform surface reconstruction from planar sketches is proposed. A multilayer perceptron (MLP) neural network is employed to induce 3D freeform surfaces from planar freehand curves. Planar curves were used to represent the boundaries of a freeform surface patch. The curves were varied iteratively and sampled to produce training data to train and test the neural network. The obtained results demonstrate that the network successfully learned the inverse-projection map and correctly inferred the respective surfaces from fresh curves

A Statistical Perspective on Randomized Sketching for Ordinary Least-Squares

We consider statistical as well as algorithmic aspects of solving large-scale least-squares (LS) problems using randomized sketching algorithms. For a LS problem with input data $(X, Y) \in \mathbb{R}^{n \times p} \times \mathbb{R}^n$, sketching algorithms use a sketching matrix, $S\in\mathbb{R}^{r \times n}$ with $r \ll n$. Then, rather than solving the LS problem using the full data $(X,Y)$, sketching algorithms solve the LS problem using only the sketched data $(SX, SY)$. Prior work has typically adopted an algorithmic perspective, in that it has made no statistical assumptions on the input $X$ and $Y$, and instead it has been assumed that the data $(X,Y)$ are fixed and worst-case (WC). Prior results show that, when using sketching matrices such as random projections and leverage-score sampling algorithms, with $p < r \ll n$, the WC error is the same as solving the original problem, up to a small constant. From a statistical perspective, we typically consider the mean-squared error performance of randomized sketching algorithms, when data $(X, Y)$ are generated according to a statistical model $Y = X \beta + \epsilon$, where $\epsilon$ is a noise process. We provide a rigorous comparison of both perspectives leading to insights on how they differ. To do this, we first develop a framework for assessing algorithmic and statistical aspects of randomized sketching methods. We then consider the statistical prediction efficiency (PE) and the statistical residual efficiency (RE) of the sketched LS estimator; and we use our framework to provide upper bounds for several types of random projection and random sampling sketching algorithms. Among other results, we show that the RE can be upper bounded when $p < r \ll n$ while the PE typically requires the sample size $r$ to be substantially larger. Lower bounds developed in subsequent results show that our upper bounds on PE can not be improved.Comment: 27 pages, 5 figure

AMS Without 4-Wise Independence on Product Domains

In their seminal work, Alon, Matias, and Szegedy introduced several sketching techniques, including showing that 4-wise independence is sufficient to obtain good approximations of the second frequency moment. In this work, we show that their sketching technique can be extended to product domains $[n]^k$ by using the product of 4-wise independent functions on $[n]$. Our work extends that of Indyk and McGregor, who showed the result for $k = 2$. Their primary motivation was the problem of identifying correlations in data streams. In their model, a stream of pairs $(i,j) \in [n]^2$ arrive, giving a joint distribution $(X,Y)$, and they find approximation algorithms for how close the joint distribution is to the product of the marginal distributions under various metrics, which naturally corresponds to how close $X$ and $Y$ are to being independent. By using our technique, we obtain a new result for the problem of approximating the $\ell_2$ distance between the joint distribution and the product of the marginal distributions for $k$-ary vectors, instead of just pairs, in a single pass. Our analysis gives a randomized algorithm that is a $(1 \pm \epsilon)$ approximation (with probability $1-\delta$) that requires space logarithmic in $n$ and $m$ and proportional to $3^k$

Sketching for Large-Scale Learning of Mixture Models

Learning parameters from voluminous data can be prohibitive in terms of memory and computational requirements. We propose a "compressive learning" framework where we estimate model parameters from a sketch of the training data. This sketch is a collection of generalized moments of the underlying probability distribution of the data. It can be computed in a single pass on the training set, and is easily computable on streams or distributed datasets. The proposed framework shares similarities with compressive sensing, which aims at drastically reducing the dimension of high-dimensional signals while preserving the ability to reconstruct them. To perform the estimation task, we derive an iterative algorithm analogous to sparse reconstruction algorithms in the context of linear inverse problems. We exemplify our framework with the compressive estimation of a Gaussian Mixture Model (GMM), providing heuristics on the choice of the sketching procedure and theoretical guarantees of reconstruction. We experimentally show on synthetic data that the proposed algorithm yields results comparable to the classical Expectation-Maximization (EM) technique while requiring significantly less memory and fewer computations when the number of database elements is large. We further demonstrate the potential of the approach on real large-scale data (over 10 8 training samples) for the task of model-based speaker verification. Finally, we draw some connections between the proposed framework and approximate Hilbert space embedding of probability distributions using random features. We show that the proposed sketching operator can be seen as an innovative method to design translation-invariant kernels adapted to the analysis of GMMs. We also use this theoretical framework to derive information preservation guarantees, in the spirit of infinite-dimensional compressive sensing

Algorithmic Perception of Vertices in Sketched Drawings of Polyhedral Shapes

In this article, visual perception principles were used to build an artificial perception model aimed at developing an algorithm for detecting junctions in line drawings of polyhedral objects that are vectorized from hand-drawn sketches. The detection is performed in two dimensions (2D), before any 3D model is available and minimal information about the shape depicted by the sketch is used. The goal of this approach is to not only detect junctions in careful sketches created by skilled engineers and designers but also detect junctions when skilled people draw casually to quickly convey rough ideas. Current approaches for extracting junctions from digital images are mostly incomplete, as they simply merge endpoints that are near each other, thus ignoring the fact that different vertices may be represented by different (but close) junctions and that the endpoints of lines that depict edges that share a common vertex may not necessarily be close to each other, particularly in quickly sketched drawings. We describe and validate a new algorithm that uses these perceptual findings to merge tips of line segments into 2D junctions that are assumed to depict 3D vertices