PRW can be seen as the generalization of Max-SW since PRW with k =1 is equivalent to Max-SW. Similar to Max-SW, the optimization of PRW is solved by using projected gradient ascent. The detailed of the algorithm is given in Algorithm 4. We would like to recall that other methods of optimization have also been used to solved PRW such as Riemannian optimization [28], block coordinate descent [21]. However, in this paper, we consider the original and simplest method which is projected gradient ascent.

The upper bound of gradient's Frobenius norm for spectrally-normalized discriminators follows directly. As lw(x) is a linear transformation, we have lcw(x) = c lw(x), and lw(cx) = c lw(x). Moreover, since ReLU and leaky ReLU is linear in R+ and R region, we have ai(cx) = c ai(x). In this section we discuss the gradients with respect the actual parameter wi. From Eq. (12) in [30] we know wtDθ(x) = A, we know that w0tDθ(x) F, otl(x)Dθ(x), and kotl (x)k have upper bounds. From Theorem 1.1 in [44] we know that if wt is initialized with i.i.d random variables from uniform or Gaussian distribution, E kwtkspis lower bounded away from zero at initialization. So k wtDθ(x)kF is upper bounded at initialization. Moreover, we observe empirically that kwtksp is usually increasing during training. Therefore, k wtDθ(x)kF is typically upper bounded during training as well. The following proposition states that spectral normalization also gives an upper bound on kHwi(Dθ)(x)ksp for networks with ReLU or leaky ReLU internal activations.

Compared computation LA-SWtoGA-SWandNA-SW, the slightly; memory usingsliced computational LA-SWis itcoststhe Section 3A-SW.

Using matrices, =U UT, where isadiagonal U is whosecolumns .

Equivariant representation learning aims to capture variations induced by input transformations in the representation space, whereas invariant representation learning encodes semantic information by disregarding such transformations. Recent studies have shown that jointly learning both types of representations is often beneficial for downstream tasks, typically by employing separate projection heads. However, this design overlooks information shared between invariant and equivariant learning, which leads to redundant feature learning and inefficient use of model capacity. To address this, we introduce Soft Task-Aware Routing (STAR), a routing strategy for projection heads that models them as experts. STAR induces the experts to specialize in capturing either shared or task-specific information, thereby reducing redundant feature learning. We validate this effect by observing lower canonical correlations between invariant and equivariant embeddings. Experimental results show consistent improvements across diverse transfer learning tasks. The code is available at https://github.com/YonseiML/star.

Non-transferable learning (NTL) has been proposed to protect model intellectual property (IP) by creating a "non-transferable barrier" to restrict generalization from authorized to unauthorized domains. Recently, well-designed attack, which restores the unauthorized-domain performance by fine-tuning NTL models on few authorized samples, highlights the security risks of NTL-based applications. However, such attack requires modifying model weights, thus being invalid in the black-box scenario. This raises a critical question: can we trust the security of NTL models deployed as black-box systems? In this work, we reveal the first loophole of black-box NTL models by proposing a novel attack method (dubbed as JailNTL) to jailbreak the non-transferable barrier through test-time data disguising. The main idea of JailNTL is to disguise unauthorized data so it can be identified as authorized by the NTL model, thereby bypassing the non-transferable barrier without modifying the NTL model weights. Specifically, JailNTL encourages unauthorized-domain disguising in two levels, including: (i) data-intrinsic disguising (DID) for eliminating domain discrepancy and preserving class-related content at the input-level, and (ii) model-guided disguising (MGD) for mitigating output-level statistics difference of the NTL model. Empirically, when attacking state-of-the-art (SOTA) NTL models in the black-box scenario, JailNTL achieves an accuracy increase of up to 55.7% in the unauthorized domain by using only 1% authorized samples, largely exceeding existing SOTA white-box attacks.

Recently, the cross-modal pretraining model has been employed to produce meaningful pseudo-labels to supervise the training of an image clustering model. However, numerous erroneous alignments in a cross-modal pre-training model could produce poor-quality pseudo-labels and degrade clustering performance. To solve the aforementioned issue, we propose a novel \textbf{Multi-level Cross-modal Alignment} method to improve the alignments in a cross-modal pretraining model for downstream tasks, by building a smaller but better semantic space and aligning the images and texts in three levels, i.e., instance-level, prototype-level, and semantic-level. Theoretical results show that our proposed method converges, and suggests effective means to reduce the expected clustering risk of our method. Experimental results on five benchmark datasets clearly show the superiority of our new method.