mathematical optimization, the convex conjugate of a function is a generalization of the Legendre transformation which applies to non-convex functions. It is also...

function. Both LogSumExp and softmax are used in machine learning. The convex conjugate (specifically, the Legendre transform) of the softplus function is...

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...

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...

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...

The convex conjugate of an extended real-valued function f : X → [ − ∞ , ∞ ] {\displaystyle f:X\to [-\infty ,\infty ]} (not necessarily convex) is the...

which identifies equivalent dynamical systems Convex conjugate, the ("dual") lower-semicontinuous convex function resulting from the Legendre–Fenchel transformation...

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...

nonnegative matrix is a convex function of its diagonal elements. Concave function Convex analysis Convex conjugate Convex curve Convex optimization Geodesic...

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...

is called the convex conjugate function of f {\displaystyle f} . For historical reasons (rooted in analytic mechanics), the conjugate variable is often...

} 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 ∗ (...

a set of points is convex if it contains every line segment between two points in the set. Equivalently, a convex set or a convex region is a set that...

{\displaystyle M} . If f {\displaystyle f} is self-concordant, then its convex conjugate f ∗ {\displaystyle f^{*}} is also self-concordant. If f {\displaystyle...

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...

example, conjugate diameters have the property that a tangent line to the ellipse at the endpoint of one diameter is parallel to the conjugate diameter...

{\displaystyle f^{*}} denotes the convex conjugate of f {\displaystyle f} . Since the subdifferential of a proper, convex, lower semicontinuous function...

In mathematics, the conjugate gradient method is an algorithm for the numerical solution of particular systems of linear equations, namely those whose...

Ψ Q ∗ {\displaystyle \Psi _{Q}^{*}} is the rate function, i.e. the convex conjugate of the cumulant-generating function, of Q {\displaystyle Q} , and μ...

negative entropy function, in physics interpreted as free entropy) is the convex conjugate of LogSumExp (in physics interpreted as the free energy). In 1953,...

Accordingly, the negative entropy (negentropy) function is convex, and its convex conjugate is LogSumExp. The inspiration for adopting the word entropy...

_{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...

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...

and concave impressions. Convex body - a compact convex set in a Euclidean space whose interior is non-empty. Convex conjugate - a dual of a real functional...

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...

_{y^{*}\in Y^{*}}[-F^{*}(0,y^{*})]} where F ∗ {\displaystyle F^{*}} is the convex conjugate in both variables. In computational optimization, another "duality...

Convex optimization is a subfield of mathematical optimization that studies the problem of minimizing convex functions over convex sets (or, equivalently...

order, for the convex conjugate function. Fixing an exponential family with log-normalizer ⁠ A {\displaystyle A} ⁠ (with convex conjugate ⁠ A ∗ {\displaystyle...

function is the convex conjugate of the cost function. Convex analysis is the standard tool for analyzing textbook economics. Non‑convex phenomena in economics...

y^{*})\leq \inf _{x\in X}F(x,0),} where F ∗ {\displaystyle F^{*}} is the convex conjugate in both variables. For any choice of perturbation function F weak duality...