Chebyshev filterChebyshev filters are analog or digital filters that have a steeper roll-off than Butterworth filters, and have either passband ripple (type I) or stopband ripple (type II). Chebyshev filters have the property that they minimize the error between the idealized and the actual filter characteristic over the operating frequency range of the filter, but they achieve this with ripples in the passband. This type of filter is named after Pafnuty Chebyshev because its mathematical characteristics are derived from Chebyshev polynomials.
Laplacian matrixIn the mathematical field of graph theory, the Laplacian matrix, also called the graph Laplacian, admittance matrix, Kirchhoff matrix or discrete Laplacian, is a matrix representation of a graph. Named after Pierre-Simon Laplace, the graph Laplacian matrix can be viewed as a matrix form of the negative discrete Laplace operator on a graph approximating the negative continuous Laplacian obtained by the finite difference method. The Laplacian matrix relates to many useful properties of a graph.
Cayley graphIn mathematics, a Cayley graph, also known as a Cayley color graph, Cayley diagram, group diagram, or color group, is a graph that encodes the abstract structure of a group. Its definition is suggested by Cayley's theorem (named after Arthur Cayley), and uses a specified set of generators for the group. It is a central tool in combinatorial and geometric group theory. The structure and symmetry of Cayley graphs makes them particularly good candidates for constructing families of expander graphs.
Electronic filter topologyElectronic filter topology defines electronic filter circuits without taking note of the values of the components used but only the manner in which those components are connected. Filter design characterises filter circuits primarily by their transfer function rather than their topology. Transfer functions may be linear or nonlinear. Common types of linear filter transfer function are; high-pass, low-pass, bandpass, band-reject or notch and all-pass.
Nonlinear dimensionality reductionNonlinear dimensionality reduction, also known as manifold learning, refers to various related techniques that aim to project high-dimensional data onto lower-dimensional latent manifolds, with the goal of either visualizing the data in the low-dimensional space, or learning the mapping (either from the high-dimensional space to the low-dimensional embedding or vice versa) itself. The techniques described below can be understood as generalizations of linear decomposition methods used for dimensionality reduction, such as singular value decomposition and principal component analysis.
Diffusion modelIn machine learning, diffusion models, also known as diffusion probabilistic models or score-based generative models, are a class of latent variable models. They are Markov chains trained using variational inference. The goal of diffusion models is to learn the latent structure of a dataset by modeling the way in which data points diffuse through the latent space. In computer vision, this means that a neural network is trained to denoise images blurred with Gaussian noise by learning to reverse the diffusion process.
Flow networkIn graph theory, a flow network (also known as a transportation network) is a directed graph where each edge has a capacity and each edge receives a flow. The amount of flow on an edge cannot exceed the capacity of the edge. Often in operations research, a directed graph is called a network, the vertices are called nodes and the edges are called arcs. A flow must satisfy the restriction that the amount of flow into a node equals the amount of flow out of it, unless it is a source, which has only outgoing flow, or sink, which has only incoming flow.
Self-organizing mapA self-organizing map (SOM) or self-organizing feature map (SOFM) is an unsupervised machine learning technique used to produce a low-dimensional (typically two-dimensional) representation of a higher dimensional data set while preserving the topological structure of the data. For example, a data set with variables measured in observations could be represented as clusters of observations with similar values for the variables.
Network synthesis filtersIn signal processing, network synthesis filters are filters designed by the network synthesis method. The method has produced several important classes of filter including the Butterworth filter, the Chebyshev filter and the Elliptic filter. It was originally intended to be applied to the design of passive linear analogue filters but its results can also be applied to implementations in active filters and digital filters. The essence of the method is to obtain the component values of the filter from a given rational function representing the desired transfer function.
Analogue filterAnalogue filters are a basic building block of signal processing much used in electronics. Amongst their many applications are the separation of an audio signal before application to bass, mid-range, and tweeter loudspeakers; the combining and later separation of multiple telephone conversations onto a single channel; the selection of a chosen radio station in a radio receiver and rejection of others.
Convolutional neural networkConvolutional neural network (CNN) is a regularized type of feed-forward neural network that learns feature engineering by itself via filters (or kernel) optimization. Vanishing gradients and exploding gradients, seen during backpropagation in earlier neural networks, are prevented by using regularized weights over fewer connections. For example, for each neuron in the fully-connected layer 10,000 weights would be required for processing an image sized 100 × 100 pixels.
Rook's graphIn graph theory, a rook's graph is an undirected graph that represents all legal moves of the rook chess piece on a chessboard. Each vertex of a rook's graph represents a square on a chessboard, and there is an edge between any two squares sharing a row (rank) or column (file), the squares that a rook can move between. These graphs can be constructed for chessboards of any rectangular shape.
