Convolution plays an important role in theory and application of mathematics. Typical convolutions include discrete convolution and continuous convolution and have close relations with Fourier transform. The concept of convolution could be extended to series, measure theory and functionals when replacing the measures. In this thesis, we introduce the convolution ideas in the studies of mixed volumes, Neural Networks and convolution algebra.

Mixed Volumes, defined on convex bodies, is related to Alexandrov-Fenchel inequality, one of the most fundamental results in convex geometry. The convolution with respect to Euler-Schanuel integral leads to a new perspective of mixed volumes. Following the convolution idea, we extend mixed volumes to the vector space spanned by the indicato...[ Read more ]

Convolution plays an important role in theory and application of mathematics. Typical convolutions include discrete convolution and continuous convolution and have close relations with Fourier transform. The concept of convolution could be extended to series, measure theory and functionals when replacing the measures. In this thesis, we introduce the convolution ideas in the studies of mixed volumes, Neural Networks and convolution algebra.

Mixed Volumes, defined on convex bodies, is related to Alexandrov-Fenchel inequality, one of the most fundamental results in convex geometry. The convolution with respect to Euler-Schanuel integral leads to a new perspective of mixed volumes. Following the convolution idea, we extend mixed volumes to the vector space spanned by the indicator functions of bounded semi-algebraic sets.

Deep Learning is a popular topic in recent years. Convolutional Neural Network has achieved much success in computer vision tasks. The interpretability of CNN remains an open problem. For CNN with ReLU as activation function, considering the role of convolution, we suggest that the activation status of the neurons are important and propose randomized methods to study the phenomenon. Numerical results are shown and in some extent support the points.

Polynomial multiplication is a discrete convolution. Following the same way complex algebra could be generalized. For a complete lattice L and a relational structure 𝔛 = (X, (R_i)_I), a new algebra L^𝔛 called convolution algebra is constructed. By setting f_i(α₁,...,α_ni)(𝓍) = V｛α₁(𝓍₁) ∧⋯∧α_ni(𝓍_ni) : (𝓍₁,…,𝓍_ni,𝓍) ∈ R_i)｝for α₁,...,α_ni ∈ L^X and 𝓍 ∈ X, the algebra L^𝔛 consists of the lattice L^X equipped with an n_i-ary operation f_i for each (n_i+1)-ary relation R_i of 𝔛. This construction is bifunctorial and extensions are made through meets instead of joins.