Linear Size Meshes
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An algorithm for producing a linear size superset of a point set that yields a linear size Delaunay triangulation in any dimension. This talk was presented at CCCG 2008.
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Supervised learning is a category of machine learning that uses labeled datasets to train algorithms to predict outcomes and recognize patterns. Unlike unsupervised learning, supervised learning algorithms are given labeled training to learn the relationship between the input and the outputs.
Supervised machine learning algorithms make it easier for organizations to create complex models that can make accurate predictions. As a result, they are widely used across various industries and fields, including healthcare, marketing, financial services, and more.
Here, we’ll cover the fundamentals of supervised learning in AI, how supervised learning algorithms work, and some of its most common use cases.
Get started for free
How does supervised learning work?
The data used in supervised learning is labeled — meaning that it contains examples of both inputs (called features) and correct outputs (labels). The algorithms analyze a large dataset of these training pairs to infer what a desired output value would be when asked to make a prediction on new data.
For instance, let’s pretend you want to teach a model to identify pictures of trees. You provide a labeled dataset that contains many different examples of types of trees and the names of each species. You let the algorithm try to define what set of characteristics belongs to each tree based on the labeled outputs. You can then test the model by showing it a tree picture and asking it to guess what species it is. If the model provides an incorrect answer, you can continue training it and adjusting its parameters with more examples to improve its accuracy and minimize errors.
Once the model has been trained and tested, you can use it to make predictions on unknown data based on the previous knowledge it has learned.
How does supervised learning work?
The data used in supervised learning is labeled — meaning that it contains examples of both inputs (called features) and correct outputs (labels). The algorithms analyze a large dataset of these training pairs to infer what a desired output value would be when asked to make a prediction on new data.
For instance, let’s pretend you want to teach a model to identify pictures of trees. You provide a labeled dataset that contains many different examples of types of trees and the names of each species. You let the algorithm try to define what set of characteristics belongs to each tree based on the labeled outputs. You can then test the model by showing it a tree picture and asking it to guess what species it is. If the model provides an incorrect answer, you can continue training it and adjusting its parameters with more examples to improve its accuracy and minimize errors.
Once the model has been trained and tested, you can use it to make predictions on unknown data based on the previous knowledge it has learned.
Types of supervised learning
Supervised learning in machine learning is generally divided into two categories: classification and regre
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clustering tendency
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Data Mining Lecture_10(b).pptx
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More from Don Sheehy
Some Thoughts on Sampling
In this talk, I address two new ideas in sampling geometric objects. The first is a new take on adaptive sampling with respect to the local feature size, i.e., the distance to the medial axis. We recently proved that such samples acn be viewed as uniform samples with respect to an alternative metric on the Euclidean space. The second is a generalization of Voronoi refinement sampling. There, one also achieves an adaptive sample while simultaneously "discovering" the underlying sizing function. We show how to construct such samples that are spaced uniformly with respect to the $k$th nearest neighbor distance function.
Characterizing the Distortion of Some Simple Euclidean Embeddings
This talk addresses some upper and lower bounds techniques for bounding the distortion between mappings between Euclidean metric spaces including circles, spheres, pairs of lines, triples of planes, and the union of a hyperplane and a point.
Sensors and Samples: A Homological Approach
In their seminal work on homological sensor networks, de Silva and Ghrist showed the surprising fact that its possible to certify the coverage of a coordinate free sensor network even with very minimal knowledge of the space to be covered. We give a new, simpler proof of the de Silva-Ghrist Topological Coverage Criterion that eliminates any assumptions about the smoothness of the boundary of the underlying space, allowing the results to be applied to much more general problems. The new proof factors the geometric, topological, and combinatorial aspects of this approach. This factoring reveals an interesting new connection between the topological coverage condition and the notion of weak feature size in geometric sampling theory. We then apply this connection to the problem of showing that for a given scale, if one knows the number of connected components and the distance to the boundary, one can also infer the higher betti numbers or provide strong evidence that more samples are needed. This is in contrast to previous work which merely assumed a good sample and gives no guarantees if the sampling condition is not met.
Persistent Homology and Nested Dissection
The document discusses using nested dissection and geometric separators to speed up computations of persistent homology. It proposes combining mesh filtrations, geometric separators, nested dissection, and output-sensitive persistence algorithms. This would allow beating the matrix multiplication time bound for computing persistent homology of functions defined on well-spaced point clouds. The technique exploits properties of meshes and separators to allow choosing a pivot order that improves the computation time.
The Persistent Homology of Distance Functions under Random Projection
Given n points P in a Euclidean space, the Johnson-Lindenstrauss lemma guarantees that the distances between pairs of points is preserved up to a small constant factor with high probability by random projection into O(log n) dimensions. In this paper, we show that the persistent homology of the distance function to P is also preserved up to a comparable constant factor. One could never hope to preserve the distance function to P pointwise, but we show that it is preserved sufficiently at the critical points of the distance function to guarantee similar persistent homology. We prove these results in the more general setting of weighted k-th nearest neighbor distances, for which k=1 and all weights equal to zero gives the usual distance to P.
Geometric and Topological Data Analysis
In this talk, I give a gentle introduction to geometric and topological data analysis and then segue into some natural questions that arise when one combines the topological view with the perhaps more well-studied linear algebraic view.
Geometric Separators and the Parabolic Lift
We present a simplification of the geometric separator algorithm of Miller and Thurston that uses parabolic lifting rather than stereographic projection. The result entirely eliminates the middle phase of that algorithm, which finds a conformal transformation to arrange the points nicely on the sphere.
A New Approach to Output-Sensitive Voronoi Diagrams and Delaunay Triangulations
We describe a new algorithm for computing the Voronoi diagram of a set of $n$ points in constant-dimensional Euclidean space. The running time of our algorithm is $O(f \log n \log \spread)$ where $f$ is the output complexity of the Voronoi diagram and $\spread$ is the spread of the input, the ratio of largest to smallest pairwise distances. Despite the simplicity of the algorithm and its analysis, it improves on the state of the art for all inputs with polynomial spread and near-linear output size. The key idea is to first build the Voronoi diagram of a superset of the input points using ideas from Voronoi refinement mesh generation. Then, the extra points are removed in a straightforward way that allows the total work to be bounded in terms of the output complexity, yielding the output sensitive bound. The removal only involves local flips and is inspired by kinetic data structures.
Optimal Meshing
The word optimal is used in different ways in mesh generation. It could mean that the output is in some sense, "the best mesh" or that the algorithm is, by some measure, "the best algorithm". One might hope that the best algorithm also produces the best mesh, but maybe some tradeoffs are necessary. In this talk, I will survey several different notions of optimality in mesh generation and explore the different tradeoffs between them. The bias will be towards Delaunay/Voronoi methods.
Output-Sensitive Voronoi Diagrams and Delaunay Triangulations
Voronoi diagrams and their duals, Delaunay triangulations, are used in many areas of computing and the sciences. Starting in 3-dimensions, there is a substantial (i.e. polynomial) difference between the best case and the worst case complexity of these objects when starting with n points. This motivates the search for algorithms that are output-senstiive rather than relying only on worst-case guarantees. In this talk, I will describe a simple, new algorithm for computing Voronoi diagrams in d-dimensions that runs in O(f log n log spread) time, where f is the output size and the spread of the input points is the ratio of the diameter to the closest pair distance. For a wide range of inputs, this is the best known algorithm. The algorithm is novel in the that it turns the classic algorithm of Delaunay refinement for mesh generation on its head, working backwards from a quality mesh to the Delaunay triangulation of the input. Along the way, we will see instances of several other classic problems for which no higher-dimensional results are known, including kinetic convex hulls and splitting Delaunay triangulations.
Mesh Generation and Topological Data Analysis
The document discusses mesh generation as a preprocessing step for topological data analysis (TDA). It describes how mesh generation can be used to decompose a domain into simple elements to approximate functions and compute persistence diagrams. Specifically, generating a quality Voronoi mesh allows the Voronoi filtration to approximate the sublevel filtration of the function and provide a good approximation of the persistence diagram. While meshing may not seem like an obvious approach for TDA, the document argues it can provide the necessary geometric and topological guarantees to make it a valid preprocessing step.
SOCG: Linear-Size Approximations to the Vietoris-Rips Filtration
The document describes the Vietoris-Rips filtration, which encodes the topology of a metric space when viewed at different scales. It introduces two tricks to create a linear-size approximation of the Vietoris-Rips filtration: 1) embedding the zigzag filtration in a topologically equivalent standard filtration, and 2) perturbing the metric so that the persistence module does not zigzag. The result is that given a metric space with n points, there exists a zigzag filtration of size O(n) whose persistence diagram approximates that of the Rips filtration. It then describes how to construct this approximating zigzag filtration using net trees and projecting the metric space onto a Delone set.
Linear-Size Approximations to the Vietoris-Rips Filtration - Presented at Uni...
This document describes a method for approximating the Vietoris-Rips filtration of a finite metric space using a zigzag filtration. The method involves two key steps: 1) embedding the zigzag filtration in a topologically equivalent standard filtration, and 2) perturbing the metric so that the persistence module does not zigzag at the homology level. The result is that given a metric space with n points, there exists a zigzag filtration of size O(n) whose persistence diagram (1+ε)-approximates that of the Rips filtration. This provides a linear-size approximation to compute topological persistence for large data sets.
Minimax Rates for Homology Inference
Often, high dimensional data lie
close to a low-dimensional submanifold and it is of interest to understand the geometry of these submanifolds.
The homology groups of a manifold are important topological invariants that provide an algebraic summary of the manifold.
These groups contain rich topological information, for instance, about the connected components, holes, tunnels and sometimes the dimension of the manifold.
In this paper, we consider the statistical problem of estimating the homology of a manifold from noisy samples under several different noise models.
We derive upper and lower bounds on the minimax risk for this problem.
Our upper bounds are based on estimators which are constructed from a union of balls of appropriate radius around carefully selected points.
In each case we establish complementary lower bounds using Le Cam's lemma.
A Multicover Nerve for Geometric Inference
We show that filtering the barycentric decomposition of a Cech complex by the cardinality of the vertices captures precisely the topology of k-covered regions among a collection of balls for all values of k.
Moreover, we relate this result to the Vietoris-Rips complex to get an approximation in terms of the persistent homology.
ATMCS: Linear-Size Approximations to the Vietoris-Rips Filtration
1) The paper presents a method to approximate the Vietoris-Rips filtration using a zigzag filtration of linear size.
2) It embeds the zigzag filtration into an equivalent standard filtration and perturbs the metric so that the zigzag does not occur at the homology level.
3) This results in a zigzag filtration of size O(n) whose persistence diagram provides a (1+ε)-approximation of the persistence diagram for the Vietoris-Rips filtration.
New Bounds on the Size of Optimal Meshes
The document discusses mesh generation, which involves decomposing a domain into simple elements like triangles or tetrahedra. An optimal mesh has good element quality, conforms to the input domain, and uses the minimum number of points needed to make all Voronoi cells sufficiently "fat" or well-shaped according to metrics like radius-edge ratios. The talk presents analysis showing that the optimal mesh size is determined by the "feature size measure" of the input points, which involves the distance to each point's second nearest neighbor.
Flips in Computational Geometry
In this talk, we will be looking at a basic primitive in computational geometry, the flip. Also known as bistellar flips, edge-flips, rotations, and Pachner moves, this local change operation has been discovered and rediscovered in a variety of fields (thus the many names) and has proven useful both as an algorithmic tool as well as a proof technology. For algorithm designers working outside of computational geometry, one can consider the flip move as a higher dimensional analog of the tree rotations used in binary trees. I will survey some of the most important results about flips with an emphasis on developing a general geometric intuition that has led to many advances.
Beating the Spread: Time-Optimal Point Meshing
We present NetMesh, a new algorithm that produces a conforming Delaunay mesh for point sets in any fixed dimension with guaranteed optimal mesh size and quality.
Our comparison based algorithm runs in time $O(n\log n + m)$, where $n$ is the input size and $m$ is the output size, and with constants depending only on the dimension and the desired element quality bounds.
It can terminate early in $O(n\log n)$ time returning a $O(n)$ size Voronoi diagram of a superset of $P$ with a relaxed quality bound, which again matches the known lower bounds.
The previous best results in the comparison model depended on the log of the <b>spread</b> of the input, the ratio of the largest to smallest pairwise distance among input points.
We reduce this dependence to $O(\log n)$ by using a sequence of $\epsilon$-nets to determine input insertion order in an incremental Voronoi diagram.
We generate a hierarchy of well-spaced meshes and use these to show that the complexity of the Voronoi diagram stays linear in the number of points throughout the construction.
Ball Packings and Fat Voronoi Diagrams
Here's a toy problem: What is the SMALLEST number of unit balls you can fit in a box such that no more will fit?
In this talk, I will show how just thinking about a naive greedy approach to this problem leads to a simple derivation of several of the most important theoretical results in the field of mesh generation.
We'll prove classic upper and lower bounds on both the number of balls and the complexity of their interrelationships.
Then, we'll relate this problem to a similar one called the Fat Voronoi Problem, in which we try to find point sets such that every Voronoi cell is fat
(the ratio of the radii of the largest contained to smallest containing ball is bounded).
This problem has tremendous promise in the future of mesh generation as it can circumvent the classic lowerbounds presented in the first half of the talk.
Unfortunately the simple approach no longer works.
In the end we will show that the number of neighbors of any cell in a Fat Voronoi Diagram in the plane is bounded by a constant
(if you think that's obvious, spend a minute to try to prove it).
We'll also talk a little about the higher dimensional version of the problem and its wide range of applications.
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Linear Size Meshes
- 1. Linear Size Meshes Gary Miller, Todd Phillips, Don Sheehy
- 2. The Meshing Problem Decompose a geometric domain into simplices. Recover features. Guarantee Quality.
- 3. The Meshing Problem Decompose a geometric domain into simplices. Recover features. Guarantee Quality.
- 4. Meshes Represent Functions Store f(v) for all mesh vertices v. Approximate f(x) by interpolating from vertices in simplex containing x.
- 5. Meshes Represent Functions Store f(v) for all mesh vertices v. Approximate f(x) by interpolating from vertices in simplex containing x.
- 6. Meshes Represent Functions Store f(v) for all mesh vertices v. Approximate f(x) by interpolating from vertices in simplex containing x.
- 7. Meshes for fluid flow simulation.
- 8. Quality A Quality triangle is one with radius-edge ratio bounded by a constant.
- 9. Quality A Quality triangle is one with radius-edge ratio bounded by a constant.
- 10. Output Size • Delaunay Triangulation of n input points may have O(n d/2 ) simplices.
- 11. Output Size • Delaunay Triangulation of n input points may have O(n d/2 ) simplices. • Quality meshes with m vertices have O(m) simplices.
- 12. Output Size • Delaunay Triangulation of n input points may have O(n d/2 ) simplices. • Quality meshes with m vertices have O(m) simplices. The number of points we add matters.
- 13. Size Optimality m: The size of our mesh. mOPT: The size of the smallest possible mesh achieving similar quality guarantees. Size Optimal: m = O(mOPT)
- 15. The Ruppert Bound lfs(x) = distance to second nearest vertex
- 16. The Ruppert Bound lfs(x) = distance to second nearest vertex Note: This bound is tight. [BernEppsteinGilbert94, Ruppert95, Ungor04, HudsonMillerPhillips06]
- 17. This work • An algorithm for producing linear size delaunay meshes of point sets in Rd.
- 18. This work • An algorithm for producing linear size delaunay meshes of point sets in Rd. • A new method for analyzing the size of quality meshes in terms of input size.
- 23. The local feature size at x is lfs(x) = |x - SN(x)|.
- 24. The local feature size at x is lfs(x) = |x - SN(x)|. A point x is θ-medial w.r.t S if |x - NN(x)| ≥ θ |x - SN(x)|.
- 25. The local feature size at x is lfs(x) = |x - SN(x)|. A point x is θ-medial w.r.t S if |x - NN(x)| ≥ θ |x - SN(x)|. An ordered set of points P is a well-paced extension of a point set Q if pi is θ-medial w.r.t. Q U {p1,...,pi-1}.
- 26. Well Paced Points A point x is θ-medial w.r.t S if |x - NNx| ≥ θ |x - SNx|. An ordered set of points P is a well-paced extension of a point set Q if pi is θ-medial w.r.t. Q U {p1,...,pi-1}.
- 31. Linear Cost to balance a Quadtree. [Moore95, Weiser81]
- 32. The Main Result
- 33. Proof Idea We want to prove that the amortized change in mopt as we add a θ-medial point is constant. lfs’ is the new local feature size after adding one point.
- 34. Proof Idea Key ideas to bound this integral. • Integrate over the entire space using polar coordinates. • Split the integral into two parts: the region near the the new point and the region far from the new point. • Use every trick you learned in high school.
- 35. How do we get around the Ruppert lower bound?
- 36. Linear Size Delaunay Meshes The Algorithm: • Pick a maximal well-paced extension of the bounding box. • Refine to a quality mesh. • Remaining points form small clusters. Surround them with smaller bounding boxes. • Recursively mesh the smaller bounding boxes. • Return the Delaunay triangulation of the entire point set.
- 46. Guarantees • Output size is O(n). • Bound on radius/longest edge. (No-large angles in R2)
- 47. Summary • An algorithm for producing linear size delaunay meshes of point sets in Rd. • A new method for analyzing the size of quality meshes in terms of input size.
- 48. Sneak Preview In a follow-up paper, we show how the theory of well-paced points leads to a proof of a classic folk conjecture. • Suppose we are meshing a domain with an odd shaped boundary. • We enclose the domain in a bounding box and mesh the whole thing. • We throw away the “extra.” • The amount we throw away is at most a constant fraction.
- 49. Thank you.
- 50. Thank you. Questions?