Learning As the Evolution of Representation

Cover Learning As the Evolution of Representation
Learning As the Evolution of Representation
Pasquale Caianiello
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Will now quant* the extra information needed.
Theorem 1 shows that there is an equivalence between C' and 5^, so we can indifferently consider any of the two. Let Ie5^, ^(l) = c. -C The additiona^ information to determine which symbol c, stands for is given by z(c. ). Its expected value is -c 7==c, o c\ ceC Then we have to multiply that value for {sf, the expec^ted cost of the codeword ^{s). We finally get that the information per symbol of T is i(r^) + l^sf {Cf (3. 6) We ofT^ only need to mult
...iply it by N' to get the expected cost of an optimal encoding 0PT-{T^) = N' (z(r^) + {Sf' (cf) ^^-^^ Now, using 3. 1 we get OPr"(T^) = ^i(I) + (^)^(^)^ (S) t' 53 (3. 8) We have proved the following: Theorem 4 Let B = S U C and T be an alphabet and a text over an alphabet A. The compression coefficient of B for T is \r{B) = -^ + {Cf The reader can get an intuition about how the classification allows a further gain in compression by considering that it causes a decrease in the number of symbols of the alphabet that is being constructed.

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