There are some cases where we have a complete description for the local optima for an optimization problem. This is a case of such an optimization problem.
Such optimization problems are useful for AI safety since a loss/fitness function where we have a complete description of all local or global optima is a highly interpretable loss/fitness function, and so one should consider using these loss/fitness functions to construct AI algorithms.
Theorem: Suppose that U is a real,complex, or quaternionic n×n-matrix that minimizes the quantity ∥U∥2+∥U−1∥2. Then U is unitary.
Proof: The real case is a special case of a complex case, and by representing each n×n-quaternionic matrix as a complex 2n×2n-matrix, we may assume that U is a complex matrix.
By the Schur decomposition, we know that U=VTV∗ where V is a unitary matrix and T is upper triangular. But we know that ∥U∥2=∥T∥2. Furthermore, U−1=VT−1V∗, so ∥U−1∥2=∥T−1∥2. Let D denote the diagonal matrix whose diagonal entries are the same as T. Then ∥T∥2≥∥D∥2 and ∥T−1∥2≥∥D−1∥2. Furthermore, ∥T∥2=∥D∥2 iff T is diagonal and ∥T−1∥2=∥D−1∥2 iff D is diagonal. Therefore, since ∥U∥2+∥U−1∥2=∥T∥2+∥T−1∥2 and ∥T∥2+∥T−1∥2 is minimized, we can conclude that T=D, so T is a diagonal matrix. Suppose that T has diagonal entries (z1,…,zn). By the arithmetic-geometric mean equality and the Cauchy-Schwarz inequality, we know that 12⋅(∥(z1,…,zn)∥2+∥(z−11,…,z−1n)∥2)≥∥(|z1|,…,|zn|)∥2⋅∥(|z−11|,…,|z−1n)|∥2
≥⟨(|z1|,…,|zn|),(|z−11|,…,|z−1n)|⟩=√n.
Here, the equalities hold if and only if |zj|=1 for all j, but this implies that U is unitary. Q.E.D.
There are some cases where we have a complete description for the local optima for an optimization problem. This is a case of such an optimization problem.
Such optimization problems are useful for AI safety since a loss/fitness function where we have a complete description of all local or global optima is a highly interpretable loss/fitness function, and so one should consider using these loss/fitness functions to construct AI algorithms.
Theorem: Suppose that U is a real,complex, or quaternionic n×n-matrix that minimizes the quantity ∥U∥2+∥U−1∥2. Then U is unitary.
Proof: The real case is a special case of a complex case, and by representing each n×n-quaternionic matrix as a complex 2n×2n-matrix, we may assume that U is a complex matrix.
By the Schur decomposition, we know that U=VTV∗ where V is a unitary matrix and T is upper triangular. But we know that ∥U∥2=∥T∥2. Furthermore, U−1=VT−1V∗, so ∥U−1∥2=∥T−1∥2. Let D denote the diagonal matrix whose diagonal entries are the same as T. Then ∥T∥2≥∥D∥2 and ∥T−1∥2≥∥D−1∥2. Furthermore, ∥T∥2=∥D∥2 iff T is diagonal and ∥T−1∥2=∥D−1∥2 iff D is diagonal. Therefore, since ∥U∥2+∥U−1∥2=∥T∥2+∥T−1∥2 and ∥T∥2+∥T−1∥2 is minimized, we can conclude that T=D, so T is a diagonal matrix. Suppose that T has diagonal entries (z1,…,zn). By the arithmetic-geometric mean equality and the Cauchy-Schwarz inequality, we know that 12⋅(∥(z1,…,zn)∥2+∥(z−11,…,z−1n)∥2)≥∥(|z1|,…,|zn|)∥2⋅∥(|z−11|,…,|z−1n)|∥2
≥⟨(|z1|,…,|zn|),(|z−11|,…,|z−1n)|⟩=√n.
Here, the equalities hold if and only if |zj|=1 for all j, but this implies that U is unitary. Q.E.D.