Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely: Consider a class $\mathcal{H}$ of binary functions h: X?{???1,?+?1} on an interval $X=[0, B]\subset \mbox{\rm IR}$ . Define the sample width of h on a finite subset (a sample) S???X as ? S (h)?=? min x???S |? h (x)| where ? h (x)?=?h(x) max {a???0: h(z)?=?h(x), x???a???z???x?+?a}. Let $\mathbb{S}_\ell$ be the space of all samples in X of cardinality ? and consider sets of wide samples, i.e., hypersets which are defined as $A_{\beta, h} = \{S\in \mathbb{S}_\ell: \omega_{S}(h) \geq \beta\}. $ Through an application of the Sauer-Shelah result on the density of sets an upper estimate is obtained on the growth function (or trace) of the class $\{A_{\beta, h}: h\in\mathcal{H}\}$ , β?>?0, i.e., on the number of possible dichotomies obtained by intersecting all hypersets with a fixed collection of samples $S\in\mathbb{S}_\ell$ of cardinality m. The estimate is $2\sum_{i=0}^{2\lfloor B/(2\beta)\rfloor}{m-\ell\choose i}$ .
['Joel Ratsaby']
I am writing a research paper on a idea On the complexity of binary samples
[{"from": "human", "value": "I am writing a research paper on a idea On the complexity of binary samples"}, {"from": "gpt", "value": "Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely: Consider a class $\\mathcal{H}$ of binary functions h: X?{???1,?+?1} on an interval $X=[0, B]\\subset \\mbox{\\rm IR}$ . Define the sample width of h on a finite subset (a sample) S???X as ? S (h)?=? min x???S |? h (x)| where ? h (x)?=?h(x) max {a???0: h(z)?=?h(x), x???a???z???x?+?a}. Let $\\mathbb{S}_\\ell$ be the space of all samples in X of cardinality ? and consider sets of wide samples, i.e., hypersets which are defined as $A_{\\beta, h} = \\{S\\in \\mathbb{S}_\\ell: \\omega_{S}(h) \\geq \\beta\\}. $ Through an application of the Sauer-Shelah result on the density of sets an upper estimate is obtained on the growth function (or trace) of the class $\\{A_{\\beta, h}: h\\in\\mathcal{H}\\}$ , \u03b2?>?0, i.e., on the number of possible dichotomies obtained by intersecting all hypersets with a fixed collection of samples $S\\in\\mathbb{S}_\\ell$ of cardinality m. The estimate is $2\\sum_{i=0}^{2\\lfloor B/(2\\beta)\\rfloor}{m-\\ell\\choose i}$ ."}]