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Laplacian eigenmaps perform non-linear dimensionality reduction ("manifold learning") by embedding data points using the eigenvectors of a graph Laplacian as coordinates.

Gabriel Peyré

102,371 subscribers

67,799 次观看 • 2 年前 •via X (Twitter)

教育科学技术

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I wrote this fun interactive article explaining Isomap, a classic non-linear dimensionality reduction technique. Isomap captures the local similarity structure of data through a graph and computes an embedding that preserves this structure. Check it out! Link below.

I wrote this fun interactive article explaining Isomap, a classic non-linear dimensionality reduction technique. Isomap captures the local similarity structure of data through a graph and computes an embedding that preserves this structure. Check it out! Link below.

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Many dimensionality reduction algorithms share a few central principles. 1. Construct a graph that captures the data's local structure 2. Measure "geodesic" distances between points using the graph 3. Project the points to a lower dimension while preserving these distances

Many dimensionality reduction algorithms share a few central principles. 1. Construct a graph that captures the data's local structure 2. Measure "geodesic" distances between points using the graph 3. Project the points to a lower dimension while preserving these distances

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#cellbiology #bioart The actin #cytoskeleton (green) is linked to a variety of transmembrane proteins through alpha-actinin (magenta), with such linkages being particularly dense in membrane ruffles and filopodia. As seen at 3 frames/min for 8 min by linear and non-linear structured illumination microscopy (97 and 60 nm resolution, respectively).

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Amazon’s machine learning model collects 300 million data points per season and can now predict which players are likely to blitz before the snap. This is a look into the future of broadcasting.

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WalkTheDog Cross-Morphology Motion Alignment via Phase Manifolds We present a new approach for understanding the periodicity structure and semantics of motion datasets, independently of the morphology and skeletal structure of characters. Unlike existing methods using an overly sparse high-dimensional latent, we propose a phase manifold consisting of multiple closed curves, each corresponding to a latent amplitude. With our proposed vector quantized periodic autoencoder, we learn a shared phase manifold for multiple characters, such as a human and a dog, without any supervision. This is achieved by exploiting the discrete structure and a shallow network as bottlenecks, such that semantically similar motions are clustered into the same curve of the manifold, and the motions within the same component are aligned temporally by the phase variable. In combination with an improved motion matching framework, we demonstrate the manifold's capability of timing and semantics alignment in several applications, including motion retrieval, transfer and stylization.

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How can we address the scarcity of data required for specialized AI? Learn about Simula, a framework that reframes synthetic data generation as dataset-level mechanism design. By using reasoning to architect datasets from first principles, Simula enables fine-grained control over coverage, complexity, and quality. More →

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The Thomas attractor is a mathematical concept in dynamical systems theory representing a type of chaotic space phase that exhibits cyclic symmetry. It was named after René Thomas, the mathematician who introduced it in 1979. The system is strictly related to the Lorenz attractor another famous chaotic evolutionary space manifold. The 3D Thomas attractor is represented by the combination of three differential equations where the symmetry is invariant under rotations of 120° (or 2π/3 radians) around the x y z axis which are the space coordinates, plus μ (mu) that is a parameter controlling the behavior. The attractor is characterized by a chaotic tendency and it’s sensitive to initial conditions, meaning that as μ increases, the motion undergoes period-doubling bifurcations leading to chaos. A good visualization of this spacetime manifold is a twisted, spiral-shaped structure with 3-fold symmetry where each spiral arm is symmetric with respect to the central axis, like the one displayed below. The chaotic evolution of this attractor demonstrates the geometric beauty and complexity of unpredictable systems.

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Our recent experiments showed us how data diversity beats quantity by lengths. A non-actuated movable base is coming soon, but until then we‘re just creating diversity by moving the table into every room of our office.

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