K-Deep Simplex：基于局部字典的深度流形学习
K-Deep Simplex: Deep Manifold Learning via Local Dictionaries
AI总结：
提出K-Deep Simplex深度流形学习方法，利用局部加权$\ell_1$惩罚学习合成地标字典与单纯形表示系数，通过算法展开构建高效可解释的自动编码器，理论证明其与加权$\ell_0$程序等价且具稳定性，实验验证了其高效性与竞争力。
摘要：
我们提出了K-Deep Simplex（KDS），给定一组数据点，它学习一个包含合成地标（landmarks）的字典，以及在一个单纯形上支持的表示系数。KDS采用局部加权$\ell_1$惩罚，鼓励每个数据点将其自身表示为附近地标的凸组合。我们使用交替最小化求解所提出的优化程序，并利用算法展开设计了一个高效、可解释的自动编码器。我们通过将KDS中的加权$\ell_1$惩罚与加权$\ell_0$程序联系起来，对所提出的程序进行了理论分析。假设数据由Delaunay三角剖分生成，我们证明了加权$\ell_1$和加权$\ell_0$程序的等价性。我们进一步展示了在温和的几何假设下表示系数的稳定性。如果表示系数固定，我们证明了在字典上最小化的子问题产生唯一解。此外，我们展示了可以从系数矩阵的协方差中高效获得低维表示。实验表明，该算法效率极高，并在合成和真实数据集上表现良好。
英文摘要：
We propose K-Deep Simplex(KDS) which, given a set of data points, learns a dictionary comprising synthetic landmarks, along with representation coefficients supported on a simplex. KDS employs a local weighted $\ell_1$ penalty that encourages each data point to represent itself as a convex combination of nearby landmarks. We solve the proposed optimization program using alternating minimization and design an efficient, interpretable autoencoder using algorithm unrolling. We theoretically analyze the proposed program by relating the weighted $\ell_1$ penalty in KDS to a weighted $\ell_0$ program. Assuming that the data are generated from a Delaunay triangulation, we prove the equivalence of the weighted $\ell_1$ and weighted $\ell_0$ programs. We further show the stability of the representation coefficients under mild geometrical assumptions. If the representation coefficients are fixed, we prove that the sub-problem of minimizing over the dictionary yields a unique solution. Further, we show that low-dimensional representations can be efficiently obtained from the covariance of the coefficient matrix. Experiments show that the algorithm is highly efficient and performs competitively on synthetic and real data sets.
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