• mathematical optimization, the convex conjugate of a function is a generalization of the Legendre transformation which applies to non-convex functions. It is also...
    16 KB (2,012 words) - 04:27, 13 May 2025
  • Thumbnail for Softplus
    function. Both LogSumExp and softmax are used in machine learning. The convex conjugate (specifically, the Legendre transform) of the softplus function is...
    5 KB (719 words) - 17:51, 2 July 2025
  • Thumbnail for Young's inequality for products
    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...
    13 KB (2,340 words) - 00:29, 7 July 2025
  • Thumbnail for Convex hull
    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
  • {\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...
    3 KB (407 words) - 14:29, 14 December 2024
  • 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...
    1 KB (208 words) - 03:46, 27 February 2023
  • Thumbnail for Convex analysis
    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
  • Thumbnail for Binary entropy function
    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
  • Thumbnail for Legendre transformation
    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
  • Thumbnail for Convex function
    nonnegative matrix is a convex function of its diagonal elements. Concave function Convex analysis Convex conjugate Convex curve Convex optimization Geodesic...
    35 KB (5,856 words) - 19:37, 21 May 2025
  • 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
  • Thumbnail for Conjugate gradient method
    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
  • Thumbnail for Entropy (information theory)
    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
  • Thumbnail for Convex set
    (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 ∗ (...
    5 KB (703 words) - 03:07, 20 April 2025
  • 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...
    23 KB (3,992 words) - 03:25, 12 April 2025
  • 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...
    26 KB (4,475 words) - 08:02, 12 January 2025
  • {\displaystyle f^{*}} denotes the convex conjugate of f {\displaystyle f} . Since the subdifferential of a proper, convex, lower semicontinuous function...
    4 KB (683 words) - 16:52, 18 January 2025
  • _{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...
    32 KB (5,094 words) - 18:38, 17 July 2025
  • negative entropy function, in physics interpreted as free entropy) is the convex conjugate of LogSumExp (in physics interpreted as the free energy). In 1953,...
    11 KB (1,225 words) - 18:26, 10 June 2025
  • order, for the convex conjugate function. Fixing an exponential family with log-normalizer ⁠ A {\displaystyle A} ⁠ (with convex conjugate ⁠ A ∗ {\displaystyle...
    86 KB (11,203 words) - 07:49, 17 July 2025
  • 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...
    28 KB (3,941 words) - 03:46, 30 June 2025
  • Thumbnail for Mathematical optimization
    with locally Lipschitz functions, particularly for convex minimization problems (similar to conjugate gradient methods). Ellipsoid method: An iterative...
    53 KB (6,155 words) - 14:53, 3 July 2025
  • 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...
    13 KB (1,850 words) - 16:00, 27 May 2025
  • Convex optimization is a subfield of mathematical optimization that studies the problem of minimizing convex functions over convex sets (or, equivalently...
    30 KB (3,170 words) - 11:17, 22 June 2025
  • assumptions on the function f {\displaystyle f} (for example, f {\displaystyle f} convex and ∇ f {\displaystyle \nabla f} Lipschitz) and particular choices of η...
    39 KB (5,600 words) - 19:08, 15 July 2025
  • Thumbnail for Dirichlet distribution
    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