Lemma 2.[Schur-Horn Lemmal Let c (ci}e Rm and b (bi}E Rm be real vectors in non-increasing order respectively,i.e.,C1≥c2≥…≥cm,b1≥b2≥.·≥bm-There exists a Hermitian matrix H with eigenvalues c and diagonal values b if and only if ∑bi≤c,for anyk=1,2,,m, =1 =1 m m ∑bi-∑ca i=1 i=1 Proof.Please refer to Horn's article [11]. 口 Base on Lemma 2,we have the following Theorem 2. Theorem 2.There exists a solution to the IsoHash problem in (3).And this solution is in the intersection ofT(a)and M(A). Pro0 f.Because≥2≥…≥and a1=a2=…=am=4,itiseasy to prove that-L≥for any.Hence,.∑1X=k×2L≥k×L= m Furthermore,we can prove that According to Lemma 2,there exists a Hermitian matrix H with eigenvalues A and diagonal values a. Moreover,we can prove that H is in the intersection of T(a)and M(A),i.e.,H E T(a)and H∈M(A). According to Theorem 2,to find a Q solving the problem in(3)is equivalent to finding the intersec- tion point of T(a)and M(A),which is just an inverse eigenvalue problem called SHIEP in [2]. 2.3 Learning The problem in(3)can be reformulated as the following optimization problem: argmin T-ZlF. (4) Q:T∈T(a),Z∈M(A) As in [2],we propose two algorithms to learn Q:one is called lift and projection(LP),and the other is called gradient flow (GF).For ease of understanding,we use the same notations as those in [2], and some proofs of theorems are omitted.The readers can refer to [2]for the details. 2.3.1 Lift and Projection The main idea of lift and projection(LP)algorithm is to alternate between the following two steps: 。Lift step: Given a T(k)E T(a),we find the point ()M(A)such thatT(k)(= dist(T(),M(A)),where dist(T(k),M(A))denotes the minimum distance between T() and the points in M(A). ·Projection step: Given a Z(),we find T(+)ET(a)such that(+)(=dist(T(a),()). where dist(T(a),Z())denotes the minimum distance between Z(k)and the points in T(a). Please note in [2].the values are in increasing order.It is easy to prove that our presentation of Schur-Horn lemma is equivalent to that in [2].The non-increasing order is chosen here just because it will facilitate our following presentation due to the non-increasing order of the eigenvalues in A.Lemma 2. [Schur-Horn Lemma] Let c = {ci} ∈ R m and b = {bi} ∈ R m be real vectors in non-increasing order respectively 1 , i.e., c1 ≥ c2 ≥ · · · ≥ cm, b1 ≥ b2 ≥ · · · ≥ bm. There exists a Hermitian matrix H with eigenvalues c and diagonal values b if and only if X k i=1 bi ≤ X k i=1 ci , for any k = 1, 2, ..., m, Xm i=1 bi = Xm i=1 ci . Proof. Please refer to Horn’s article [11]. Base on Lemma 2, we have the following Theorem 2. Theorem 2. There exists a solution to the IsoHash problem in (3). And this solution is in the intersection of T (a) and M(Λ). Proof. Because λ1 ≥ λ2 ≥ · · · ≥ λm and a1 = a2 = · · · = am = Pm i=1 λi m , it is easy to prove that Pk i=1 λi k ≥ Pm i=1 λi m for any k. Hence, Pk i=1 λi = k × Pk i=1 λi k ≥ k × Pm i=1 λi m = Pk i=1 ai . Furthermore, we can prove that Pm i=1 λi = Pm i=1 ai . According to Lemma 2, there exists a Hermitian matrix H with eigenvalues λ and diagonal values a. Moreover, we can prove that H is in the intersection of T (a) and M(Λ), i.e., H ∈ T (a) and H ∈ M(Λ). According to Theorem 2, to find a Q solving the problem in (3) is equivalent to finding the intersec￾tion point of T (a) and M(Λ), which is just an inverse eigenvalue problem called SHIEP in [2]. 2.3 Learning The problem in (3) can be reformulated as the following optimization problem: argmin Q:T ∈T (a),Z∈M(Λ) ||T − Z||F . (4) As in [2], we propose two algorithms to learn Q: one is called lift and projection (LP), and the other is called gradient flow (GF). For ease of understanding, we use the same notations as those in [2], and some proofs of theorems are omitted. The readers can refer to [2] for the details. 2.3.1 Lift and Projection The main idea of lift and projection (LP) algorithm is to alternate between the following two steps: • Lift step: Given a T (k) ∈ T (a), we find the point Z (k) ∈ M(Λ) such that ||T (k) − Z (k) ||F = dist(T (k) ,M(Λ)), where dist(T (k) ,M(Λ)) denotes the minimum distance between T (k) and the points in M(Λ). • Projection step: Given a Z (k) , we find T (k+1) ∈ T (a) such that ||T (k+1) − Z (k) ||F = dist(T (a), Z(k) ), where dist(T (a), Z(k) ) denotes the minimum distance between Z (k) and the points in T (a). 1 Please note in [2], the values are in increasing order. It is easy to prove that our presentation of Schur-Horn lemma is equivalent to that in [2]. The non-increasing order is chosen here just because it will facilitate our following presentation due to the non-increasing order of the eigenvalues in Λ. 4