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...

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

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...

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

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

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...

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...

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

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

} 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 ∗ (...

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

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...

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...

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...

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...

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

_{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...

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

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...

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...

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

generally, any convex polyhedron or cone has a polyhedral dual, and any convex set X with boundary hypersurface H has a convex conjugate X* whose boundary...

divergence to a Bregman divergence is the divergence generated by the convex conjugate F* of the Bregman generator of the original divergence. For example...

assumptions on the function F {\displaystyle F} (for example, F {\displaystyle F} convex and ∇ F {\displaystyle \nabla F} Lipschitz) and particular choices of γ...

{\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...

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

used to solve non-differentiable convex optimization problems. Many interesting problems can be formulated as convex optimization problems of the form...