mathematical optimization, the convex conjugate of a function is a generalization of the Legendre transformation which applies to non-convex functions. It is also...
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Softplus (section Convex conjugate)
function. Both LogSumExp and softmax are used in machine learning. The convex conjugate (specifically, the Legendre transform) of the softplus function is...
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version for conjugate Hölder exponents. For details and generalizations we refer to the paper of Mitroi & Niculescu. By denoting the convex conjugate of a real...
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In geometry, the convex hull, convex envelope or convex closure of a shape is the smallest convex set that contains it. The convex hull may be defined...
58 KB (7,173 words) - 01:04, 1 July 2025
Self-concordant function (section Convex conjugate)
{\displaystyle M} . If f {\displaystyle f} is self-concordant, then its convex conjugate f ∗ {\displaystyle f^{*}} is also self-concordant. If f {\displaystyle...
22 KB (4,403 words) - 16:59, 19 January 2025
Conjugation (redirect from Conjugate)
which identifies equivalent dynamical systems Convex conjugate, the ("dual") lower-semicontinuous convex function resulting from the Legendre–Fenchel transformation...
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spaces Convex function, when the line segment between any two points on the graph of the function lies above or on the graph Convex conjugate, of a function...
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The convex conjugate of an extended real-valued function f : X → [ − ∞ , ∞ ] {\displaystyle f:X\to [-\infty ,\infty ]} (not necessarily convex) is the...
16 KB (2,605 words) - 20:34, 8 June 2025
Binary entropy function (section Convex conjugate)
dp^{2}}\operatorname {H} _{\text{b}}(p)=-{\frac {1}{p(1-p)\ln 2}}} The convex conjugate (specifically, the Legendre transform) of the binary entropy (with...
6 KB (1,071 words) - 17:05, 6 May 2025
Legendre transformation (category Convex analysis)
is called the convex conjugate function of f {\displaystyle f} . For historical reasons (rooted in analytic mechanics), the conjugate variable is often...
51 KB (8,917 words) - 18:28, 3 July 2025
nonnegative matrix is a convex function of its diagonal elements. Concave function Convex analysis Convex conjugate Convex curve Convex optimization Geodesic...
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which means the gradient of LogSumExp is the softmax function. The convex conjugate of LogSumExp is the negative entropy. The LSE function is often encountered...
7 KB (1,152 words) - 17:21, 23 June 2024
In mathematics, the conjugate gradient method is an algorithm for the numerical solution of particular systems of linear equations, namely those whose...
51 KB (8,421 words) - 13:05, 20 June 2025
Accordingly, the negative entropy (negentropy) function is convex, and its convex conjugate is LogSumExp. The inspiration for adopting the word entropy...
71 KB (10,208 words) - 07:29, 15 July 2025
(pages 562–563): Krein, M.; Šmulian, V. (1940). "On regularly convex sets in the space conjugate to a Banach space". Annals of Mathematics. Second Series....
27 KB (3,429 words) - 17:52, 10 May 2025
Fenchel's duality theorem (category Convex optimization)
} where ƒ * is the convex conjugate of ƒ (also referred to as the Fenchel–Legendre transform) and g * is the concave conjugate of g. That is, f ∗ (...
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distances (Nielsen & Nock (2013)). Let f ∗ {\displaystyle f^{*}} be the convex conjugate of f {\displaystyle f} . Let e f f d o m ( f ∗ ) {\displaystyle \mathrm...
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Duality: If F is strictly convex, then the function F has a convex conjugate F ∗ {\displaystyle F^{*}} which is also strictly convex and continuously differentiable...
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{\displaystyle f^{*}} denotes the convex conjugate of f {\displaystyle f} . Since the subdifferential of a proper, convex, lower semicontinuous function...
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_{t}at-K(t)} The moment generating function is log-convex, so by a property of the convex conjugate, the Chernoff bound must be log-concave. The Chernoff...
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negative entropy function, in physics interpreted as free entropy) is the convex conjugate of LogSumExp (in physics interpreted as the free energy). In 1953,...
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order, for the convex conjugate function. Fixing an exponential family with log-normalizer A {\displaystyle A} (with convex conjugate A ∗ {\displaystyle...
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Duality (optimization) (category Convex optimization)
y^{*})\leq \inf _{x\in X}F(x,0),\,} where F ∗ {\displaystyle F^{*}} is the convex conjugate in both variables and sup {\displaystyle \sup } denotes the supremum...
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with locally Lipschitz functions, particularly for convex minimization problems (similar to conjugate gradient methods). Ellipsoid method: An iterative...
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divergence to a Bregman divergence is the divergence generated by the convex conjugate F* of the Bregman generator of the original divergence. For example...
20 KB (2,629 words) - 14:02, 17 June 2025
{\displaystyle \Psi _{Q}^{*}} is the large deviations rate function, i.e. the convex conjugate of the cumulant-generating function, of Q, and μ 1 ′ ( P ) {\displaystyle...
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Convex optimization is a subfield of mathematical optimization that studies the problem of minimizing convex functions over convex sets (or, equivalently...
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assumptions on the function f {\displaystyle f} (for example, f {\displaystyle f} convex and ∇ f {\displaystyle \nabla f} Lipschitz) and particular choices of η...
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Dirichlet distribution (category Conjugate prior distributions)
the moment-generating function of the Dirichlet distribution to the convex conjugate of the scaled reversed Kullback-Leibler divergence: log E ( exp...
49 KB (7,756 words) - 03:37, 9 July 2025
used to solve non-differentiable convex optimization problems. Many interesting problems can be formulated as convex optimization problems of the form...
5 KB (589 words) - 12:28, 21 June 2025