LogSumExp

The LogSumExp function domain is \R^n, the real coordinate space, and its codomain is \R, the real line. It is an approximation to the maximum \max_i x_i with the following bounds \max{\{x_1, \dots, x_n\}} \leq \mathrm{LSE}(x_1, \dots, x_n) \leq \max{\{x_1, \dots, x_n\}} + \log(n). The first inequality is strict unless n = 1. The second inequality is strict unless all arguments are equal. (Proof: Let m = \max_i x_i. Then \exp(m) \leq \sum_{i=1}^n \exp(x_i) \leq n \exp(m). Applying the logarithm to the inequality gives the result.) In addition, we can scale the function to make the bounds tighter. Consider the function \frac 1 t \mathrm{LSE}(tx_1, \dots, tx_n). Then \max{\{x_1, \dots, x_n\}} (Proof: Replace each x_i with tx_i for some t>0 in the inequalities above, to give \max{\{tx_1, \dots, tx_n\}} and, since t>0 t \max{\{x_1, \dots, x_n\}} finally, dividing by t gives the result.) Also, if we multiply by a negative number instead, we of course find a comparison to the \min function: \min{\{x_1, \dots, x_n\}} - \frac{\log(n)}{t} \leq \frac 1 {-t} \mathrm{LSE}(-tx) The LogSumExp function is convex, and is strictly increasing everywhere in its domain. It is not strictly convex, since it is affine (linear plus a constant) on the diagonal and parallel lines: :\mathrm{LSE}(x_1 + c, \dots, x_n + c) =\mathrm{LSE}(x_1, \dots, x_n) + c. Other than this direction, it is strictly convex (the Hessian has rank ), so for example restricting to a hyperplane that is transverse to the diagonal results in a strictly convex function. See \mathrm{LSE}_0^+, below. Writing \mathbf{x} = (x_1, \dots, x_n), the partial derivatives are: \frac{\partial}{\partial x_i}{\mathrm{LSE}(\mathbf{x})} = \frac{\exp x_i}{\sum_j \exp {x_j}}, which means the gradient of LogSumExp is the softmax function. The convex conjugate of LogSumExp is the negative entropy. ==log-sum-exp trick for log-domain calculations==

The LSE function is often encountered when the usual arithmetic computations are performed on a logarithmic scale, as in log probability. Similar to multiplication operations in linear-scale becoming simple additions in log-scale, an addition operation in linear-scale becomes the LSE in log-scale: \mathrm{LSE}(\log(x_1), ..., \log(x_n)) = \log(x_1 + \dots + x_n) A common purpose of using log-domain computations is to increase accuracy and avoid underflow and overflow problems when very small or very large numbers are represented directly (i.e. in a linear domain) using limited-precision floating point numbers. Unfortunately, the use of LSE directly in this case can again cause overflow/underflow problems. Therefore, the following equivalent must be used instead (especially when the accuracy of the above 'max' approximation is not sufficient). \mathrm{LSE}(x_1, \dots, x_n) = x^* + \log\left( \exp(x_1-x^*)+ \cdots + \exp(x_n-x^*) \right) where x^* = \max{\{x_1, \dots, x_n\}} Many math libraries such as IT++ provide a default routine of LSE and use this formula internally. == A strictly convex log-sum-exp type function ==

LSE is convex but not strictly convex. We can define a strictly convex log-sum-exp type function by adding an extra argument set to zero: \mathrm{LSE}_0^+(x_1,...,x_n) = \mathrm{LSE}(0,x_1,...,x_n) This function is a proper Bregman generator (strictly convex and differentiable). It is encountered in machine learning, for example, as the cumulant of the multinomial/binomial family. In tropical analysis, this is the sum in the log semiring. == See also ==