In image and audio signal classification, a major problem is to build stable representations that are invariant under rigid motions and, more generally, to small diffeomorphisms. Translation invariant representations of signals in ${\mathbb{C}}^n$ are of particular importance. The existence of such representations is intimately related to classical invariant theory, inverse problems in compressed sensing and deep learning. Despite an impressive body of literature on the subject, most representations available are either not stable due to the presence of high frequencies or non discriminative. In the present paper, we construct low dimensional representations of signals in ${\mathbb{C}}^n$ that are invariant under finite unitary group actions, as a special case we establish the existence of low-dimensional and complete ${\mathbb{Z}}_m$-invariant representations for any $m\in \mathbb{N}$. Our construction yields a stable, discriminative transform with semi-explicit Lipschitz bounds on the dimension; this is particularly relevant for applications. Using some tools from Algebraic Geometry, we define a high dimensional homogeneous function that is injective. We then exploit the projective character of this embedding and see that the target space can be reduced significantly by using a generic linear transformation. Finally, we introduce the notion of non-parallel map, which is enjoyed by our function and employ this to construct a Lipschitz modification of it.

在图像和音频信号分类中，一个主要问题是要建立在刚体运动和更普遍的情况下对于小微同态不变的稳定表示。信号的平移不变表示${\mathbb{C}}^{ñ}$特别重要。此类表示的存在与经典不变理论，压缩感测和深度学习中的逆问题密切相关。尽管有关该主题的文献令人印象深刻，但由于存在高频，大多数可用的表示方法要么不稳定，要么没有歧视性。在本文中，我们构造了信号中的低维表示${\mathbb{C}}^{ñ}$ 在有限的group群作用下是不变的，作为一种特殊情况，我们确定了低维和完整的存在 ${\mathbb{ž}}_{米}$-任何的不变表示 $米\in \mathbb{ñ}$。我们的构造产生了一个稳定的，判别性的变换，在维度上具有半明显的Lipschitz边界；这对于应用特别重要。使用“代数几何”中的一些工具，我们定义了一个高维齐射的齐次函数。然后，我们利用该嵌入的投影特性，发现可以通过使用通用线性变换显着减小目标空间。最后，我们介绍了非并行映射的概念，它是我们的功能所享受的，并以此构建了Lipschitz的修改。