H. Ogawa, "Theory of pseudo biorthogonal bases and its application," Research Institute for Mathematical Science, RIMS Kokyuroku, vol. 1067, Reproducing Kernels and their Applications, pp. 24--38, 1998.  Home/Search   Document Not in Database   Summary   Related Articles   Check

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H. Ogawa, "Theory of pseudo biorthogonal bases and its application," Research Institute for Mathematical Science, RIMS Kokyuroku, vol. 1067, Reproducing Kernels and their Applications, pp. 24--38, 1998.

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H. Ogawa, "Theory of pseudo biorthogonal bases and its application," Research Institute for Mathematical Science, RIMS Kokyuroku, No. 1067, Reproducing Kernels and their Applications, pp. 24--38, 1998.

....and Iijima [20, 21] presented the concept of POBs in 1973, independently of the Du#n and Schae#er s work. A POB is a normalized tight frame in the frame terminology. Ogawa [12, 13] extended POBs to pseudo biorthogonal bases (PBOBs) in 1978, and devoted himself to showing their properties in detail [14, 19]. So far, PBOBs have been applied to many problems such as signal restoration [15] computerized tomography [22] and NN learning [18] Especially in NN learning, PBOBs play a major role when we discuss the optimal generalization capability and the robustness of NNs. One of the most important ....

H. Ogawa, Theory of pseudo biorthogonal bases and its application, in "Research Institute for Mathematical Science, RIMS Kokyuroku, no. 1067 in Reproducing Kernels and their Applications", pp. 24--38, 1998.

....and lemmas are provided in B. Active Learning for Optimal Generalization in Trigonometric Polynomial Models 6 3.2 Mechanism of achieving optimal generalization capability Eq. 21) is equivalent to that a set # rM #m M m=1 of sampling functions forms a pseudo orthonormal basis (PONB) 19][18] in H. The concept of PONBs is an extension of orthonormal bases to linearly dependent over complete systems. The rigorous definition and properties of PONBs are described in A. Using the properties of PONBs, we have the following lemma. Lemma 1 When the sampling operator A satisfies Condition ....

....as f = M # m=1 #f, #m ##m . 61) The concept of POBs is an extension of orthonormal bases (ONBs) to linearly dependent over complete systems. It is clear that a POB is reduced to an ONB in H if M is equal to the dimension of H. POBs and their extension, pseudo biorthogonal bases [15][18], have been successfully applied to various real world problems including signal restoration [16] 18] computerized tomography [20] neural network learning [17] and robust construction of neural networks [13] 11] The following proposition shows basic characteristics of POBs. Proposition 2 [19] ....

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H. Ogawa, "Theory of pseudo biorthogonal bases and its application," Research Institute for Mathematical Science, RIMS Kokyuroku, vol. 1067, Reproducing Kernels and their Applications, pp. 24--38, 1998.

....is minimized under the constraint N (A) 0 if and only if A # A = MI, 12) where I denotes the identity operator on H. In this case, the minimum value of JG is # 2 M , where is the dimension of H. Eq. 12) implies that 1 # M #m M m=1 forms a pseudo orthonormal basis (Ogawa [8]) in H , which is an extension of orthonormal bases. The following lemma gives interpretation of Theorem 1. Lemma 1 When a set xm M m=1 of sample points satisfies Eq. 12) it holds that XAf = f for all f # H, 13) #Af# = # M#f# for all f # H, 14) #Xu# = # 1 # M #u# for u # ....

H. Ogawa. Theory of pseudo biorthogonal bases and its application. In Research Institute for Mathematical Science, RIMS Kokyuroku, 1067, Reproducing Kernels and their Applications, pp. 24--38, 1998.

No context found.

H. Ogawa, "Theory of pseudo biorthogonal bases and its application," in Research Institute for Mathematical Science, RIMS Kokyuroku, No. 1067 in Reproducing Kernels and their Applications, pp. 24--38, Oct. 1998.

....is minimized under the constraint of N (A) 0 if and only if A # A = MI , 20) where I denotes the identity operator in H . In this case, the minimum value of JG is # 2 M . Eq. 20) implies that 1 # M #m M m=1 forms a pseudo orthonormal basis (Ogawa and Iijima [11] Ogawa [10]) in H , which is an extension of orthonormal bases. The following lemma gives interpretation of Theorem 1. Lemma 1 When a set xm M m=1 of sample points satisfies eq. 20) it holds that #Af# = # M#f# for all u # H, 21) XAu = u for all u # H, 22) #Xv# = # 1 # M #v# for v # ....

H. Ogawa. Theory of pseudo biorthogonal bases and its application. In Research Institute for Mathematical Science, RIMS Kokyuroku, 1067, Reproducing Kernels and their Applications, pp. 24--38, 1998.

....and Iijima [20, 21] presented the concept of POBs in 1973, independently of the Du#n and Schae#er s work. A POB is a normalized tight frame in the frame terminology. Ogawa [12, 13] extended POBs to pseudo biorthogonal bases (PBOBs) in 1978, and devoted himself to showing their properties in detail [14, 19]. So far, PBOBs have been applied to many problems such as signal restoration [15] computerized tomography [22] and NN learning [18] Especially in NN learning, PBOBs play a major role when we discuss the optimal generalization capability and the robustness of NNs. One of the most important ....

H. Ogawa, Theory of pseudo biorthogonal bases and its application, in "Research Institute for Mathematical Science, RIMS Kokyuroku, no. 1067 in Reproducing Kernels and their Applications", pp. 24--38, 1998.

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