Competitive learning

⚠️ Curated Limitations & Negative Results

• The normalized binary-input rule requires a nonempty input, a declared winner or tie convention, initialized unit-total weights and an appropriate gain. Zero input leaves x/n undefined; gains outside [0,1] need not preserve the simplex, and endpoint gains do not preserve strict positivity in general. Input normalization, gain schedules and competition rules are realization-specific, not fixed for the entire paradigm.
• The inspected reprint contains a scaling discrepancy in its equilibrium discussion. Its zero-expected-drift equation on PDF p. 21 / printed p. 168 implies, for positive learning gain, constant active count n and positive winning probability, w_i=P(input i active | unit wins)/n. The next page prints n times that conditional probability. The preceding equation and unit-total constraint support division by n; this is a discrepancy in the inspected 1986 reprint, not an asserted author erratum or a claim about uninspected 1985 journal bytes.
• Unit-total nonnegative weights have fixed L1 norm, not fixed Euclidean norm. The reprint qualifies its geometric analogy, and maximum weighted input is not universally equivalent to nearest Euclidean center under that constraint. Winner-only adaptation can leave units unused; neither equal occupancy, distinct semantic features nor a unique stable partition follows in general. Zero expected weight drift also does not itself prove convergence of continually updated random weights.
• The soft weighted-mean condition assumes a valid mixture and positive variance and total responsibility. It is a fixed-point or stationary relation because its responsibilities depend on the means. Computing responsibilities under old parameters and maximizing a fixed-responsibility surrogate can give an EM mean step; an arbitrary online gradient step, cumulative average or decayed statistic update is not automatically that procedure. No global maximum, monotone likelihood for arbitrary finite steps or unique online schedule is established here.
• Nowlan’s Equation 1 prints a prefactor 1/(K*sigma_j) with covariance sigma_j^2 I. A normalized d-dimensional spherical Gaussian requires a factor proportional to sigma_j^(-d); the printed form needs a dimension/variance-dependent interpretation of K outside one dimension. A common fixed factor cancels in the equal-prior, common-variance responsibilities used here, so this caveat does not invalidate that restricted comparison. Maximizing log likelihood calls for ascent, or descent of its negative, despite the source’s “gradient descent” wording; no general literal variance-update implementation is certified.
• The maximum component term in Nowlan’s Equation 4 is a hard approximation to the mixture sum, not generally an equal normalized density. Unequal priors or covariances change assignments and responsibilities. Hard centroid fitting, marginal-likelihood EM and annealed or stochastic competitive procedures therefore are not unconditional synonyms, and finite experimental comparisons do not establish universal superiority or convergence.
• This paradigm is not identical to Lloyd k-means, a self-organizing map, PBIL or exact-cache incremental EM. A self-organizing map additionally adapts a topological neighborhood; PBIL updates a search distribution using objective-selected candidates; exact-cache incremental EM has its own fixed-data replacement schedule. In Nowlan’s application a separately trained output layer uses class labels, so an unsupervised center-fitting component does not make the complete classifier unsupervised.
• Primary evidence uses the inspected 1986 Rumelhart–Zipser chapter reprint and official NIPS 1989 Nowlan proceedings PDF. The reprint identifies its 1985 journal origin, but journal-byte equality was not checked; reading used selected OCR and formula visuals, not the unread middle experiment pages. The official Nowlan year 1989 and historical citation year 1990 are bibliographic conventions, not two algorithms. Yuille’s 1994 note supplies contextual EM/annealing discussion only. The unavailable 1991 Nowlan thesis, author implementations and benchmark reproduction were not inspected in this research; general biological or semantic-feature guarantees are not inferred.