The swish function is a family of mathematical function defined as follows: swish β ( x ) = x sigmoid ( β x ) = x 1 + e − β x . {\displaystyle \operatorname {swish} _{\beta }(x)=x\operatorname {sigmoid} (\beta x)={\frac {x}{1+e^{-\beta x}}}.} where β {\displaystyle \beta } can be constant (usually set to 1) or trainable and "sigmoid" refers to the logistic function. The swish family was designed to smoothly interpolate between a linear function and the Rectified linear unit (ReLU) function. When considering positive values, Swish is a particular case of doubly parameterized sigmoid shrinkage function defined in . Variants of the swish function include Mish. == Special values == For β = 0, the function is linear: f(x) = x/2. For β = 1, the function is the Sigmoid Linear Unit (SiLU). For β = 1.702, the function approximates GeLU. With β → ∞, the function converges to ReLU. Thus, the swish family smoothly interpolates between a linear function and the ReLU function. Since swish β ( x ) = swish 1 ( β x ) / β {\displaystyle \operatorname {swish} _{\beta }(x)=\operatorname {swish} _{1}(\beta x)/\beta } , all instances of swish have the same shape as the default swish 1 {\displaystyle \operatorname {swish} _{1}} , zoomed by β {\displaystyle \beta } . One usually sets β > 0 {\displaystyle \beta >0} . When β {\displaystyle \beta } is trainable, this constraint can be enforced by β = e b {\displaystyle \beta =e^{b}} , where b {\displaystyle b} is trainable. swish 1 ( x ) = x 2 + x 2 4 − x 4 48 + x 6 480 + O ( x 8 ) {\displaystyle \operatorname {swish} _{1}(x)={\frac {x}{2}}+{\frac {x^{2}}{4}}-{\frac {x^{4}}{48}}+{\frac {x^{6}}{480}}+O\left(x^{8}\right)} swish 1 ( x ) = x 2 tanh ( x 2 ) + x 2 swish 1 ( x ) + swish − 1 ( x ) = x tanh ( x 2 ) swish 1 ( x ) − swish − 1 ( x ) = x {\displaystyle {\begin{aligned}\operatorname {swish} _{1}(x)&={\frac {x}{2}}\tanh \left({\frac {x}{2}}\right)+{\frac {x}{2}}\\\operatorname {swish} _{1}(x)+\operatorname {swish} _{-1}(x)&=x\tanh \left({\frac {x}{2}}\right)\\\operatorname {swish} _{1}(x)-\operatorname {swish} _{-1}(x)&=x\end{aligned}}} == Derivatives == Because swish β ( x ) = swish 1 ( β x ) / β {\displaystyle \operatorname {swish} _{\beta }(x)=\operatorname {swish} _{1}(\beta x)/\beta } , it suffices to calculate its derivatives for the default case. swish 1 ′ ( x ) = x + sinh ( x ) 4 cosh 2 ( x 2 ) + 1 2 {\displaystyle \operatorname {swish} _{1}'(x)={\frac {x+\sinh(x)}{4\cosh ^{2}\left({\frac {x}{2}}\right)}}+{\frac {1}{2}}} so swish 1 ′ ( x ) − 1 2 {\displaystyle \operatorname {swish} _{1}'(x)-{\frac {1}{2}}} is odd. swish 1 ″ ( x ) = 1 − x 2 tanh ( x 2 ) 2 cosh 2 ( x 2 ) {\displaystyle \operatorname {swish} _{1}''(x)={\frac {1-{\frac {x}{2}}\tanh \left({\frac {x}{2}}\right)}{2\cosh ^{2}\left({\frac {x}{2}}\right)}}} so swish 1 ″ ( x ) {\displaystyle \operatorname {swish} _{1}''(x)} is even. == History == SiLU was first proposed alongside the GELU in 2016, then again proposed in 2017 as the Sigmoid-weighted Linear Unit (SiL) in reinforcement learning. The SiLU/SiL was then again proposed as the SWISH over a year after its initial discovery, originally proposed without the learnable parameter β, so that β implicitly equaled 1. The swish paper was then updated to propose the activation with the learnable parameter β. In 2017, after performing analysis on ImageNet data, researchers from Google indicated that using this function as an activation function in artificial neural networks improves the performance, compared to ReLU and sigmoid functions. It is believed that one reason for the improvement is that the swish function helps alleviate the vanishing gradient problem during backpropagation.
Kolmogorov–Arnold Networks
Kolmogorov–Arnold Networks (KANs) are a type of artificial neural network architecture inspired by the Kolmogorov–Arnold representation theorem, also known as the superposition theorem. Unlike traditional multilayer perceptrons (MLPs), which rely on fixed activation functions and linear weights, KANs replace each weight with a learnable univariate function, often represented using splines. == History == KANs (Kolmogorov–Arnold Networks) were proposed by Liu et al. (2024) as a generalization of the Kolmogorov–Arnold representation theorem (KART), aiming to outperform MLPs in small-scale AI and scientific tasks. Before KANs, numerous studies explored KART's connections to neural networks or used it as a basis for designing new network architectures. In the 1980s and 1990s, early research applied KART to neural network design. Kůrková et al. (1992), Hecht-Nielsen (1987), and Nees (1994) established theoretical foundations for multilayer networks based on KART. Igelnik et al. (2003) introduced the Kolmogorov Spline Network using cubic splines to model complex functions. Sprecher (1996, 1997) introduced numerical methods for building network layers, while Nakamura et al. (1993) created activation functions with guaranteed approximation accuracy. These works linked KART's theoretical potential with practical neural network implementation. KART has also been used in other computational and theoretical fields. Coppejans (2004) developed nonparametric regression estimators using B-splines, Bryant (2008) applied it to high-dimensional image tasks, Liu (2015) investigated theoretical applications in optimal transport and image encryption, and more recently, Polar and Poluektov (2021) used Urysohn operators for efficient KART construction, while Fakhoury et al. (2022) introduced ExSpliNet, integrating KART with probabilistic trees and multivariate B-splines for improved function approximation. == Architecture == KANs are based on the Kolmogorov–Arnold representation theorem, which was linked to the 13th Hilbert problem. Given x = ( x 1 , x 2 , … , x n ) {\displaystyle x=(x_{1},x_{2},\dots ,x_{n})} consisting of n variables, a multivariate continuous function f ( x ) {\displaystyle f(x)} can be represented as: f ( x ) = f ( x 1 , … , x n ) = ∑ q = 1 2 n + 1 Φ q ( ∑ p = 1 n φ q , p ( x p ) ) {\displaystyle f(x)=f(x_{1},\dots ,x_{n})=\sum _{q=1}^{2n+1}\Phi _{q}\left(\sum _{p=1}^{n}\varphi _{q,p}(x_{p})\right)} (1) This formulation contains two nested summations: an outer and an inner sum. The outer sum ∑ q = 1 2 n + 1 {\displaystyle \sum _{q=1}^{2n+1}} aggregates 2 n + 1 {\displaystyle 2n+1} terms, each involving a function Φ q : R → R {\displaystyle \Phi _{q}:\mathbb {R} \to \mathbb {R} } . The inner sum ∑ p = 1 n {\displaystyle \sum _{p=1}^{n}} computes n terms for each q, where each term φ q , p : [ 0 , 1 ] → R {\displaystyle \varphi _{q,p}:[0,1]\to \mathbb {R} } is a continuous function of the single variable x p {\displaystyle x_{p}} . The inner continuous functions φ q , p {\displaystyle \varphi _{q,p}} are universal, independent of f {\displaystyle f} , while the outer functions Φ q {\displaystyle \Phi _{q}} depend on the specific function f {\displaystyle f} being represented. The representation (1) holds for all multivariate functions f {\displaystyle f} as proved in . If f {\displaystyle f} is continuous, then the outer functions Φ q {\displaystyle \Phi _{q}} are continuous; if f {\displaystyle f} is discontinuous, then the corresponding Φ q {\displaystyle \Phi _{q}} are generally discontinuous, while the inner functions φ q , p {\displaystyle \varphi _{q,p}} remain the same universal functions. Liu et al. proposed the name KAN. A general KAN network consisting of L layers takes x to generate the output as: K A N ( x ) = ( Φ L − 1 ∘ Φ L − 2 ∘ ⋯ ∘ Φ 1 ∘ Φ 0 ) x {\displaystyle \mathrm {KAN} (x)=(\Phi ^{L-1}\circ \Phi ^{L-2}\circ \cdots \circ \Phi ^{1}\circ \Phi ^{0})x} (3) Here, Φ l {\displaystyle \Phi ^{l}} is the function matrix of the l-th KAN layer or a set of pre-activations. Let i denote the neuron of the l-th layer and j the neuron of the (l+1)-th layer. The activation function φ j , i l {\displaystyle \varphi _{j,i}^{l}} connects (l, i) to (l+1, j): φ j , i l , l = 0 , … , L − 1 , i = 1 , … , n l , j = 1 , … , n l + 1 {\displaystyle \varphi _{j,i}^{l},\quad l=0,\dots ,L-1,\;i=1,\dots ,n_{l},\;j=1,\dots ,n_{l+1}} (4) where nl is the number of nodes of the l-th layer. Thus, the function matrix Φ l {\displaystyle \Phi ^{l}} can be represented as an n l + 1 × n l {\displaystyle n_{l+1}\times n_{l}} matrix of activations: x l + 1 = ( φ 1 , 1 l ( ⋅ ) φ 1 , 2 l ( ⋅ ) ⋯ φ 1 , n l l ( ⋅ ) φ 2 , 1 l ( ⋅ ) φ 2 , 2 l ( ⋅ ) ⋯ φ 2 , n l l ( ⋅ ) ⋮ ⋮ ⋱ ⋮ φ n l + 1 , 1 l ( ⋅ ) φ n l + 1 , 2 l ( ⋅ ) ⋯ φ n l + 1 , n l l ( ⋅ ) ) x l {\displaystyle x^{l+1}={\begin{pmatrix}\varphi _{1,1}^{l}(\cdot )&\varphi _{1,2}^{l}(\cdot )&\cdots &\varphi _{1,n_{l}}^{l}(\cdot )\\\varphi _{2,1}^{l}(\cdot )&\varphi _{2,2}^{l}(\cdot )&\cdots &\varphi _{2,n_{l}}^{l}(\cdot )\\\vdots &\vdots &\ddots &\vdots \\\varphi _{n_{l+1},1}^{l}(\cdot )&\varphi _{n_{l+1},2}^{l}(\cdot )&\cdots &\varphi _{n_{l+1},n_{l}}^{l}(\cdot )\end{pmatrix}}x^{l}} == Implementations == To make the KAN layers optimizable, the inner function is formed by the combination of spline and basic functions as the formula: φ ( x ) = w b b ( x ) + w s spline ( x ) {\displaystyle \varphi (x)=w_{b}\,b(x)+w_{s}\,{\text{spline}}(x)} where b ( x ) {\displaystyle b(x)} is the basic function, usually defined as s i l u ( x ) = x / ( 1 + e x ) {\displaystyle silu(x)=x/(1+e^{x})} and w b {\displaystyle w_{b}} is the base weight matrix. Also, w s {\displaystyle w_{s}} is the spline weight matrix and spline ( x ) {\displaystyle {\text{spline}}(x)} is the spline function. The spline function can be a sum of B-splines. spline ( x ) = ∑ i c i B i ( x ) {\displaystyle {\text{spline}}(x)=\sum _{i}c_{i}B_{i}(x)} Many studies suggested to use other polynomial and curve functions instead of B-spline to create new KAN variants. == Functions used == The choice of functional basis strongly influences the performance of KANs. Common function families include: B-splines: Provide locality, smoothness, and interpretability; they are the most widely used in current implementations. RBFs (include Gaussian RBFs): Capture localized features in data and are effective in approximating functions with non-linear or clustered structures. Chebyshev polynomials: Offer efficient approximation with minimized error in the maximum norm, making them useful for stable function representation. Rational function: Useful for approximating functions with singularities or sharp variations, as they can model asymptotic behavior better than polynomials. Fourier series: Capture periodic patterns effectively and are particularly useful in domains such as physics-informed machine learning. Wavelet functions (DoG, Mexican hat, Morlet, and Shannon): Used for feature extraction as they can capture both high-frequency and low-frequency data components. Piecewise linear functions: Provide efficient approximation for multivariate functions in KANs. == Usage == In some modern neural architectures like convolutional neural networks (CNNs), recurrent neural networks (RNNs), and Transformers, KANs are typically used as drop-in substitutes for MLP layers. Despite KANs' general-purpose design, researchers have created and used them for a number of tasks: Scientific machine learning (SciML): Function fitting, partial differential equations (PDEs) and physical/mathematical laws. Continual learning: KANs better preserve previously learned information during incremental updates, avoiding catastrophic forgetting due to the locality of spline adjustments. Graph neural networks: Extensions such as Kolmogorov–Arnold Graph Neural Networks (KA-GNNs) integrate KAN modules into message-passing architectures, showing improvements in molecular property prediction tasks. Sensor data processing: Kolmogorov–Arnold Networks (KANs) have recently been applied to sensor data processing due to their ability to model complex nonlinear relationships with relatively few parameters and improved interpretability compared to conventional multilayer perceptrons. Applications include industrial soft sensors, biomedical signal analysis, remote sensing, and environmental monitoring systems. == Drawbacks == KANs can be computationally intensive and require a large number of parameters due to their use of polynomial functions to capture data.
Vlado Keselj
Vlado Keselj (Vlado Kešelj) is a Serbian-Canadian computer scientist known for his research in natural language processing and authorship attribution. He is a professor at Dalhousie University. == Education == As a high school student in Yugoslavia, Keselj competed in the 1987 International Mathematical Olympiad, earning a bronze medal. He earned his Ph.D. in 2002 at the University of Waterloo, with the dissertation Modular Stochastic HPSGs for Question Answering supervised by Nick Cercone. == Awards == Vlado Keselj is a recipient of the 2019 CAIAC Distinguished Service Award, awarded by the Canadian Artificial Intelligence Association (CAIAC). == Selected publications == Kešelj, V., Peng, F., Cercone, N., & Thomas, C. (2003, August). N-gram-based author profiles for authorship attribution. In Proceedings of the Conference of the Pacific Association for Computational Linguistics, PACLING 2003 (Vol. 3, pp. 255–264).
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Weight initialization
In deep learning, weight initialization or parameter initialization describes the initial step in creating a neural network. A neural network contains trainable parameters that are modified during training: weight initialization is the pre-training step of assigning initial values to these parameters. The choice of weight initialization method affects the speed of convergence, the scale of neural activation within the network, the scale of gradient signals during backpropagation, and the quality of the final model. Proper initialization is necessary for avoiding issues such as vanishing and exploding gradients and activation function saturation. Note that even though this article is titled "weight initialization", both weights and biases are used in a neural network as trainable parameters, so this article describes how both of these are initialized. Similarly, trainable parameters in convolutional neural networks (CNNs) are called kernels and biases, and this article also describes these. == Constant initialization == We discuss the main methods of initialization in the context of a multilayer perceptron (MLP). Specific strategies for initializing other network architectures are discussed in later sections. For an MLP, there are only two kinds of trainable parameters, called weights and biases. Each layer l {\displaystyle l} contains a weight matrix W ( l ) ∈ R n l − 1 × n l {\displaystyle W^{(l)}\in \mathbb {R} ^{n_{l-1}\times n_{l}}} and a bias vector b ( l ) ∈ R n l {\displaystyle b^{(l)}\in \mathbb {R} ^{n_{l}}} , where n l {\displaystyle n_{l}} is the number of neurons in that layer. A weight initialization method is an algorithm for setting the initial values for W ( l ) , b ( l ) {\displaystyle W^{(l)},b^{(l)}} for each layer l {\displaystyle l} . The simplest form is zero initialization: W ( l ) = 0 , b ( l ) = 0 {\displaystyle W^{(l)}=0,b^{(l)}=0} Zero initialization is usually used for initializing biases, but it is not used for initializing weights, as it leads to symmetry in the network, causing all neurons to learn the same features. In this page, we assume b = 0 {\displaystyle b=0} unless otherwise stated. Recurrent neural networks typically use activation functions with bounded range, such as sigmoid and tanh, since unbounded activation may cause exploding values. (Le, Jaitly, Hinton, 2015) suggested initializing weights in the recurrent parts of the network to identity and zero bias, similar to the idea of residual connections and LSTM with no forget gate. In most cases, the biases are initialized to zero, though some situations can use a nonzero initialization. For example, in multiplicative units, such as the forget gate of LSTM, the bias can be initialized to 1 to allow good gradient signal through the gate. For neurons with ReLU activation, one can initialize the bias to a small positive value like 0.1, so that the gradient is likely nonzero at initialization, avoiding the dying ReLU problem. == Random initialization == Random initialization means sampling the weights from a normal distribution or a uniform distribution, usually independently. === LeCun initialization === LeCun initialization, popularized in (LeCun et al., 1998), is designed to preserve the variance of neural activations during the forward pass. It samples each entry in W ( l ) {\displaystyle W^{(l)}} independently from a distribution with mean 0 and variance 1 / n l − 1 {\displaystyle 1/n_{l-1}} . For example, if the distribution is a continuous uniform distribution, then the distribution is U ( ± 3 / n l − 1 ) {\displaystyle {\mathcal {U}}(\pm {\sqrt {3/n_{l-1}}})} . === Glorot initialization === Glorot initialization (or Xavier initialization) was proposed by Xavier Glorot and Yoshua Bengio. It was designed as a compromise between two goals: to preserve activation variance during the forward pass and to preserve gradient variance during the backward pass. For uniform initialization, it samples each entry in W ( l ) {\displaystyle W^{(l)}} independently and identically from U ( ± 6 / ( n l + 1 + n l − 1 ) ) {\displaystyle {\mathcal {U}}(\pm {\sqrt {6/(n_{l+1}+n_{l-1})}})} . In the context, n l − 1 {\displaystyle n_{l-1}} is also called the "fan-in", and n l + 1 {\displaystyle n_{l+1}} the "fan-out". When the fan-in and fan-out are equal, then Glorot initialization is the same as LeCun initialization. === He initialization === As Glorot initialization performs poorly for ReLU activation, He initialization (or Kaiming initialization) was proposed by Kaiming He et al. for networks with ReLU activation. It samples each entry in W ( l ) {\displaystyle W^{(l)}} from N ( 0 , 2 / n l − 1 ) {\displaystyle {\mathcal {N}}(0,2/n_{l-1})} . === Orthogonal initialization === (Saxe et al. 2013) proposed orthogonal initialization: initializing weight matrices as uniformly random (according to the Haar measure) semi-orthogonal matrices, multiplied by a factor that depends on the activation function of the layer. It was designed so that if one initializes a deep linear network this way, then its training time until convergence is independent of depth. Sampling a uniformly random semi-orthogonal matrix can be done by initializing X {\displaystyle X} by IID sampling its entries from a standard normal distribution, then calculate ( X X ⊤ ) − 1 / 2 X {\displaystyle \left(XX^{\top }\right)^{-1/2}X} or its transpose, depending on whether X {\displaystyle X} is tall or wide. For CNN kernels with odd widths and heights, orthogonal initialization is done this way: initialize the central point by a semi-orthogonal matrix, and fill the other entries with zero. As an illustration, a kernel K {\displaystyle K} of shape 3 × 3 × c × c ′ {\displaystyle 3\times 3\times c\times c'} is initialized by filling K [ 2 , 2 , : , : ] {\displaystyle K[2,2,:,:]} with the entries of a random semi-orthogonal matrix of shape c × c ′ {\displaystyle c\times c'} , and the other entries with zero. (Balduzzi et al., 2017) used it with stride 1 and zero-padding. This is sometimes called the Orthogonal Delta initialization. Related to this approach, unitary initialization proposes to parameterize the weight matrices to be unitary matrices, with the result that at initialization they are random unitary matrices (and throughout training, they remain unitary). This is found to improve long-sequence modelling in LSTM. Orthogonal initialization has been generalized to layer-sequential unit-variance (LSUV) initialization. It is a data-dependent initialization method, and can be used in convolutional neural networks. It first initializes weights of each convolution or fully connected layer with orthonormal matrices. Then, proceeding from the first to the last layer, it runs a forward pass on a random minibatch, and divides the layer's weights by the standard deviation of its output, so that its output has variance approximately 1. === Fixup initialization === In 2015, the introduction of residual connections allowed very deep neural networks to be trained, much deeper than the ~20 layers of the previous state of the art (such as the VGG-19). Residual connections gave rise to their own weight initialization problems and strategies. These are sometimes called "normalization-free" methods, since using residual connection could stabilize the training of a deep neural network so much that normalizations become unnecessary. Fixup initialization is designed specifically for networks with residual connections and without batch normalization, as follows: Initialize the classification layer and the last layer of each residual branch to 0. Initialize every other layer using a standard method (such as He initialization), and scale only the weight layers inside residual branches by L − 1 2 m − 2 {\displaystyle L^{-{\frac {1}{2m-2}}}} . Add a scalar multiplier (initialized at 1) in every branch and a scalar bias (initialized at 0) before each convolution, linear, and element-wise activation layer. Similarly, T-Fixup initialization is designed for Transformers without layer normalization. === Others === Instead of initializing all weights with random values on the order of O ( 1 / n ) {\displaystyle O(1/{\sqrt {n}})} , sparse initialization initialized only a small subset of the weights with larger random values, and the other weights zero, so that the total variance is still on the order of O ( 1 ) {\displaystyle O(1)} . Random walk initialization was designed for MLP so that during backpropagation, the L2 norm of gradient at each layer performs an unbiased random walk as one moves from the last layer to the first. Looks linear initialization was designed to allow the neural network to behave like a deep linear network at initialization, since W R e L U ( x ) − W R e L U ( − x ) = W x {\displaystyle W\;\mathrm {ReLU} (x)-W\;\mathrm {ReLU} (-x)=Wx} . It initializes a matrix W {\displaystyle W} of shape R n 2 × m {\displaystyle \mathbb {R} ^{{\frac {n}{2}}\times m}} by any method, such as orthogonal initialization, t
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