Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely: Sum of Squares Error and Normalized Cut are two widely used clustering functional. It is known their minimum values are monotone with respect to the input number of clusters and this monotonicity does not allow for a simple automatic selection of a correct number of clusters. Here  study monotonicity not just on the minimizers but on the entire clustering lattice.  show the value of Sum of Squares Error is strictly monotone under the strict refinement relation of clusterings and  obtain data-dependent bounds on the difference beten the value of a clustering and one of its refinements. Using analogous techniques  show the value of Normalized Cut is strictly anti-monotone. These results imply that even if  restrict our solutions to form a chain of clustering, like the one  get from hierarchical algorithms,  cannot rely on the functional values in order to choose the number of clusters. By using these results  get some data-dependent bounds on the difference of the values of any two clusterings.
['Nicola Rebagliati']
I am writing a research paper on a idea Strict Monotonicity of Sum of Squares Error and Normalized Cut in the Lattice of Clusterings
[{"from": "human", "value": "I am writing a research paper on a idea Strict Monotonicity of Sum of Squares Error and Normalized Cut in the Lattice of Clusterings"}, {"from": "gpt", "value": "Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely: Sum of Squares Error and Normalized Cut are two widely used clustering functional. It is known their minimum values are monotone with respect to the input number of clusters and this monotonicity does not allow for a simple automatic selection of a correct number of clusters. Here  study monotonicity not just on the minimizers but on the entire clustering lattice.  show the value of Sum of Squares Error is strictly monotone under the strict refinement relation of clusterings and  obtain data-dependent bounds on the difference beten the value of a clustering and one of its refinements. Using analogous techniques  show the value of Normalized Cut is strictly anti-monotone. These results imply that even if  restrict our solutions to form a chain of clustering, like the one  get from hierarchical algorithms,  cannot rely on the functional values in order to choose the number of clusters. By using these results  get some data-dependent bounds on the difference of the values of any two clusterings."}]