Artificial Intelligence and Quantum Computing for Advanced Wireless Networks. Savo G. Glisic

      Supervision signal for the previous layer: In the l‐th layer, we need to compute ∂z/∂vec(x^l). For that, we want to reshape x^l into a matrix , and use these two equivalent forms (modulo reshaping) interchangeably. By the chain rule, ∂z/∂(vec(x^l)^T) = [∂z/∂(vec(y)^T)][∂vec(y)/∂(vec(x^l)^T)].

By utilizing Eqs. (3.90) and (3.96), we have

      (3.97)

      (3.98)

      Since by using Eq. (3.88)

      (3.99)

      we have

      (3.100)

      or equivalently

(3.101)

In Eq. (3.101), , and vec((∂z/∂Y)F^T) is a vector in . At the same time, M^T is an indicator matrix in In order to locate one element in vec(x^l) or one row in M^T, we need an index triplet (i^l, j^l, d^l), with 0 ≤ i^l < H^l, 0 ≤ j^l < W^l, and 0 ≤ d^l < D^l. Similarly, to locate a column in M^T or an element in ∂z/∂Y)F^T, we need an index pair p ,q), with 0≤p < H^{l + 1} W^{l + 1}and ≤ q< HW D^l. Thus, the (i^l, j^l, d^l )‐th entry of ∂z/∂(vec(x^l)) is the product of two vectors: the row in M^T (or the column in M) that is indexed by (i^l, j^l, d^l), and vec((∂z/∂Y)F^T). Since M^T is an indicator matrix, in the row vector indexed by (i^l, j^l, d^l), only those entries whose index (p, q) satisfies m(p, q) = (i^l, j^l, d^l) have a value 1, and all other entries are 0. Thus, the (i^l, j^l, d^l )‐th entry of ∂z/∂(vec(x^l)) equals the sum of these corresponding entries in vec((∂z/∂Y)F^T). Therefore, we get the following succinct equation:

(3.102)

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