| # Log-Sum-Exp (LSE) Function and Properties The log-sum-exp (LSE) function in is the function , with domain the whole space , and value at a point given by
| The log-sum-exp function in . For large positive values, the function is a smooth approximation to the maximum function .
The log-sum-exp function is increasing with respect to each argument, and convex. Proof: The monotonicity of the log-sum-exp function is obvious. The convexity is obtained as follows. As seen here, the Hessian of the log-sum-exp function is where , and . We need to check that for every , we have . Let us fix a vector . We have due to the Cauchy-Schwartz inequality.