{\displaystyle \mathrm {LSE} (x_{1},\dots ,x_{n})=\log \left(\exp(x_{1})+\cdots +\exp(x_{n})\right).} The LogSumExp function domain is R n {\displaystyle \mathbb...

Boltzmann distribution. Another smooth maximum is LogSumExp: L S E α ( x 1 , … , x n ) = 1 α log ⁡ ∑ i = 1 n exp ⁡ α x i {\displaystyle \mathrm {LSE} _{\alpha...

The multivariable generalization of single-variable softplus is the LogSumExp with the first argument set to zero: L S E 0 + ⁡ ( x 1 , … , x n ) :=...

\mathbb {R} ^{K},} where the LogSumExp function is defined as LSE ⁡ ( z 1 , … , z n ) = log ⁡ ( exp ⁡ ( z 1 ) + ⋯ + exp ⁡ ( z n ) ) {\displaystyle \operatorname...

(log multiplication), and takes addition to log addition (LogSumExp), giving an isomorphism of semirings between the probability semiring and the log semiring...

then the exponential function of Y, X = exp(Y), has a log-normal distribution. A random variable which is log-normally distributed takes only positive...

The multivariable generalization of single-variable softplus is the LogSumExp with the first argument set to zero: L S E 0 + ⁡ ( x 1 , … , x n ) :=...

operation, logadd (for multiple terms, LogSumExp) can be viewed as a deformation of maximum or minimum. The log-semiring has applications in mathematical...

the sum of its Maclaurin series. The exponential satisfies the functional equation: exp ⁡ ( x + y ) = exp ⁡ ( x ) ⋅ exp ⁡ ( y ) . {\displaystyle \exp(x+y)=\exp(x)\cdot...

entropy (negentropy) function is convex, and its convex conjugate is LogSumExp. The inspiration for adopting the word entropy in information theory came...

system Logarithmic scale Logarithmic spiral Logarithmic timeline Logit LogSumExp Mantissa is a disambiguation page; see common logarithm for the traditional...

= exp ⁡ ( x ) {\displaystyle f(x)=\exp(x)} , then the f-mean is the mean in the log semiring, which is a constant shifted version of the LogSumExp (LSE)...

of max ( x 0 , x 1 ) {\displaystyle \max(x_{0},x_{1})} , specifically LogSumExp. Softplus thus generalizes as (note the 0 and the corresponding 1 for...

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

D(x^{i}-c_{\alpha }(x))\,B^{i}.} If one can employ an efficient implementation of the LogSumExp functions, this can be beneficial for numerical stability of the consensus...

has a slope of 1 at x = 0. One has e = exp ⁡ ( 1 ) , {\displaystyle e=\exp(1),} where exp {\displaystyle \exp } is the (natural) exponential function...

p − 1 exp ⁡ ( 2 π i n 2 q p ) = e π i / 4 2 q ∑ n = 0 2 q − 1 exp ⁡ ( − π i n 2 p 2 q ) {\displaystyle \displaystyle {\dfrac {1}{\sqrt {p}}}\sum _{n=0}^{p-1}\exp...

exp ⁡ ( 2 π i x ) . {\displaystyle e(x)=\exp(2\pi ix).\,} Therefore, a typical exponential sum may take the form ∑ n e ( x n ) , {\displaystyle \sum _{n}e(x_{n})...

{\displaystyle x=0.} LogSumExp function, also called softmax function, is a convex function. The function − log ⁡ det ( X ) {\displaystyle -\log \det(X)} on the...

instances of log(x) without a subscript base should be interpreted as a natural logarithm, also commonly written as ln(x) or loge(x). The sum of the reciprocals...

with the meaning of expm1(x) = exp(x) − 1. An identity in terms of the inverse hyperbolic tangent, l o g 1 p ( x ) = log ⁡ ( 1 + x ) = 2   a r t a n h...

{ad_{Z}} &=\log \left(\exp \left(\mathrm {ad} _{Z}\right)\right)=\log \left(1+\left(\exp \left(\mathrm {ad} _{Z}\right)-1\right)\right)\\&=\sum \limits _{n=1}^{\infty...

as a sum and the power as a multiplication: When a 1 , a 2 , … , a n > 0 {\displaystyle a_{1},a_{2},\dots ,a_{n}>0} ( ∏ i = 1 n a i ) 1 n = exp ⁡ ( 1...

{\displaystyle f_{i}} , i.e., the posynomials, are transformed into log-sum-exp functions, which are convex, and the functions g i {\displaystyle g_{i}}...

{\displaystyle (-t)^{z-1}} is interpreted as exp ⁡ ( ( z − 1 ) log ⁡ ( − t ) ) {\displaystyle \exp((z-1)\log(-t))} . The reflection formula leads to the...

Q ) = ∑ x ∈ X P ( x ) log ⁡ P ( x ) Q ( x ) . {\displaystyle D_{\text{KL}}(P\parallel Q)=\sum _{x\in {\mathcal {X}}}P(x)\,\log {\frac {P(x)}{Q(x)}}{\text{...

_{i}(g_{i}+x_{i}))={\frac {e^{x_{j}}}{\sum _{i}e^{x_{i}}}}} Related equations include: If x ∼ Exp ⁡ ( λ ) {\displaystyle x\sim \operatorname {Exp} (\lambda )} , then (...

constitutes a formal object, extending the field of exp-log functions of Hardy and the field of accelerando-summable series of Ecalle. The field T L E {\displaystyle...

equivalent form log 2 ⁡ ( n ! ) = n log 2 ⁡ n − n log 2 ⁡ e + O ( log 2 ⁡ n ) . {\displaystyle \log _{2}(n!)=n\log _{2}n-n\log _{2}e+O(\log _{2}n).} The...

above for the sum of differences from the mean): p ( X ∣ μ , τ ) = ∏ i = 1 n τ 2 π exp ⁡ ( − 1 2 τ ( x i − μ ) 2 ) = ( τ 2 π ) n / 2 exp ⁡ ( − 1 2 τ ∑...