This paper studies Graphical SLOPE for precision matrix estimation, with emphasis on its ability to recover both sparsity and clusters of edges with equal or similar strength. In a fixed-dimensional regime, we establish that the root-$n$ scaled estimation error converges to the unique minimizer of a strictly convex optimization problem defined through the directional derivative of the SLOPE penalty. We also establish convergence of the induced SLOPE pattern, thereby obtaining an asymptotic characterization of the clustering structure selected by the estimator. A comparison with GLASSO shows that the grouping property of SLOPE can substantially improve estimation accuracy when the precision matrix exhibits structured edge patterns. To assess the effect of departures from Gaussianity, we then analyze Gaussian-loss precision matrix estimation under elliptical distributions. In this setting, we derive the limiting distribution and quantify the inflation in variability induced by heavy tails relative to the Gaussian benchmark. We also study TSLOPE, based on the multivariate $t$-loss, and derive its limiting distribution. The results show that TSLOPE offers clear advantages over GSLOPE under heavy-tailed data-generating mechanisms. Simulation evidence suggests that these qualitative conclusions persist in high-dimensional settings, and an empirical application shows that SLOPE-based estimators, especially TSLOPE, can uncover economically meaningful clustered dependence structures.

W e are also committed to releasing the code. Implementation details for Stage 2. Our implementation strictly follows the previous work that also In this section, we briefly introduce our tasks. It requires the robot hand to open the door on the table. It requires the robot hand to orient the pen to the target orientation. It requires the robot hand to place the object on the table into the mug. We present the success rates of our six task categories as in Table 1.

Here, we include implementation details omitted from the main paper for brevity. Upon acceptance, a deanonymized repository will be released. The last layer's dimension depended upon the exact The feature extractors and decoders varied by domain. In particular, we found that if we did not apply this linear transformation (i.e., pass the raw encodings For VQ-based methods, use a large enough codebook to have at least one element per class. Other differences simply reflected differences in architecture (e.g., For iNat, we trained all models with batch size 256, using the hyperparameters specified in Table 3.

Note that we don't validate the inner-loop'sλ at every outer-loop iteration, but keep changing it on-the-fly at each validation cycle.

We generate 128,000 images as agents' observations using python's matplotlib library Hunter [2007] V ariational autoencoder [Kingma and Welling, 2014] is used to encode the observations. Input is flatted 30,720-dimensional vector (32 by 320 by 3). Both encoder and decoder have one hidden layer with the dimension size being 1,024. The output (communication message) is a 10-dimensional vector. ReLU is used as the activation function.