AI-Powered Reading Companion

2.1 Notations and Definitions2.1 Notations and Definitions

We write eᵢ := (0, …, 1, 0, …, 0)ᵀ ∈ ℝᵈ for the basis vector whose i-th component is 1. For x, y ∈ ℝᵈ, (x ∨ y)ᵢ = max{xᵢ, yᵢ} and (x ∧ y)ᵢ = min{xᵢ, yᵢ}. Given 𝒟 ⊆ ℝᵈ, a continuous f : 𝒟 → ℝ is submodular when f(x) + f(y) ≥ f(x ∨ y) + f(x ∧ y) for all x, y ∈ 𝒟.We write eᵢ := (0, …, 1, 0, …, 0)ᵀ ∈ ℝᵈ for the basis vector whose i-th component is 1. For x, y ∈ ℝᵈ, (x ∨ y)ᵢ = max{xᵢ, yᵢ} and (x ∧ y)ᵢ = min{xᵢ, yᵢ}. Given 𝒟 ⊆ ℝᵈ, a continuous f : 𝒟 → ℝ is submodular when f(x) + f(y) ≥ f(x ∨ y) + f(x ∧ y) for all x, y ∈ 𝒟.

For 𝒟 ⊆ ℝᵈ, f : 𝒟 → ℝ is L₀-Lipschitz with respect to ‖·‖ when |f(x) − f(y)| ≤ L₀‖x − y‖ for all x, y ∈ 𝒟, and L-smooth when its gradient is L-Lipschitz. If f is also twice differentiable, submodularity means every off-diagonal entry of its Hessian is non-positive.For 𝒟 ⊆ ℝᵈ, f : 𝒟 → ℝ is L₀-Lipschitz with respect to ‖·‖ when |f(x) − f(y)| ≤ L₀‖x − y‖ for all x, y ∈ 𝒟, and L-smooth when its gradient is L-Lipschitz. If f is also twice differentiable, submodularity means every off-diagonal entry of its Hessian is non-positive.

Reading context (adapted): eᵢ is the basis vector whose i-th coordinate is 1 and whose other coordinates are 0. The definition requires a to be nonnegative and both resulting points, x + aeᵢ and y + aeᵢ, to remain in the domain 𝒟. The inequality compares two changes in function value. The DR property alone does not guarantee positive increments, nor does it mean that the total function value falls as inputs grow. Keep this definition separate from additional assumptions when reading the methods that follow.Reading context (adapted): eᵢ is the basis vector whose i-th coordinate is 1 and whose other coordinates are 0. The definition requires a to be nonnegative and both resulting points, x + aeᵢ and y + aeᵢ, to remain in the domain 𝒟. The inequality compares two changes in function value. The DR property alone does not guarantee positive increments, nor does it mean that the total function value falls as inputs grow. Keep this definition separate from additional assumptions when reading the methods that follow.

Reading context (adapted): the inequality has to hold in each coordinate direction i on its own, with x + aeᵢ and y + aeᵢ both remaining in 𝒟. It compares two increments rather than two values of f, which is what “diminishing returns” refers to.Reading context (adapted): the inequality has to hold in each coordinate direction i on its own, with x + aeᵢ and y + aeᵢ both remaining in 𝒟. It compares two increments rather than two values of f, which is what “diminishing returns” refers to.