维中的

给定数据点，每个数据点具有特征，标记为，其他标记为。每个特征随机取的值（均匀分布）。存在可以分裂两个类别的超平面的概率是多少？d Ñ / 2 0 Ñ / 2 1 [ 0 ，1 $n$$d$$n/2$$0$$n/2$$1$$[0,1]$

让我们首先考虑最简单的情况，即。$d = 1$

假设数据中不存在重复项。

如果$n\leq d+1$，则概率为$\text{Pr}=1$。

对于$(n,d)$其他组合，请参见下图：

我生成了该图，以模拟OP中指定的输入和输出数据。由于Hauck-Donner效应，线性可分离性被定义为逻辑回归模型中的收敛失败。

我们可以看到增加的概率降低。实际上，我们可以拟合一个将与关联的模型结果是：$n$$n, d$$p$

情节代码（朱莉娅）：

using GLM

ds = 10; #number of dimensions to be investigated
ns = 100 #number of examples to be investigated
niter = 1000; #number of iterations per d per n
P = niter * ones(Int64, ds, ns); #starting the number of successes

for d in 1:ds
    for n in (d+1):ns
        p = 0 #0 hits
        for i in 1:niter
            println("Dimensions: $d; Samples: $n; Iteration: $i;")
            try #we will try to catch errors in the logistic glm, these are due to perfect separability
                X = hcat(rand((n,d)), ones(n)); #sampling from uniform plus intercept
                Y = sample(0:1, n)  #sampling a binary outcome
                glm(X, Y, Binomial(), LogitLink())
            catch
                p = p+1 #if we catch an error, increase the count
            end
        end
        P[d,n] = p
    end
end

using Plots

gui(heatmap(P./niter, xlabel = "Number of Samples", ylabel = "Number of Dimensions", title = "Probability of linear separability"))

有关与（在Julia中）的模型的代码：$(n,d)$$p$

probs = P./niter
N = transpose(repmat(1:ns, 1, ds))
D = repmat(1:ds, 1, ns)

fit = glm(hcat(log.(N[:]), D[:], N[:].*D[:], ones(ds*ns)), probs[:], Binomial(), LogitLink())
coef(fit)
#4-element Array{Float64,1}:
# -4.58261
#  1.37271
# -0.0235785
#  5.82944

gui(heatmap(reshape(predict(fit), ds, ns), xlabel = "Number of Samples", ylabel = "Number of Dimensions", title = "Fit of probability of linear separability"))