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For a nonconvex function $p_\lambda(\cdot)$, with hyper-parameter $\lambda$, it can be decomposed into two parts that $p_\lambda(t) = \lambda |t| + q_\lambda(t)$, where $q_\lambda(t)$ is the concave component. Particularly, $p_\lambda(t)$ and $q_\lambda(t)$ satisfy the following regularity conditions:

1. There exists a constant $\nu$ such that the function $p_{\lambda}(t)$ satisfies $p'_{\lambda}(t) = 0, \forall~ t \geq \nu > 0$.

2. On the nonnegative real line, $q'_{\lambda}(t)$ is monotone and Lipschitz continuous, i.e., for $t' \geq t$, there exists a constant $\zeta_- \geq 0$ such that $ q'_{\lambda}(t') - q'_{\lambda}(t) \geq -\zeta_- (t'-t)$.

3. Both function $q_{\lambda}(t)$ and its derivative $q'_{\lambda}(t)$ pass through the origin, i.e., $q_{\lambda}(0) = q'_{\lambda}(0) = 0$.

4. On the nonnegative real line, $|q'_{\lambda}(t)|$ is upper bounded by $\lambda$, i.e., $|q'_{\lambda}(t)| \leq \lambda$.

For a matrix $\mathbf{M} \in \mathbb{R}^{d_1 \times d_2}$, define the nonconvex function $P_\lambda(\cdot)$ on the singular value vector of $\mathbf{M}$ such that $P_\lambda(\mathbf{M}) = \sum_{i = 1}^{d} p_\lambda(\sigma_i(\mathbf{M}))$, where $d = \min\{d_1, d_2\}$ and $\sigma_i(\mathbf{M})$ is the $i$-th singular value of $\mathbf{M}$.

Based on these definitions, I am wondering if the following claim is correct or not that, for arbitrary $\mathbf{A, B} \in \mathbb{R}^{d_1 \times d_2}$, $$ P_\lambda( \mathbf{A + B}) \leq P_\lambda( \mathbf{A}) + P_\lambda( \mathbf{B}).$$

For a nonconvex function $p_\lambda(\cdot)$, with hyper-parameter $\lambda$, it can be decomposed into two parts that $p_\lambda(t) = \lambda |t| + q_\lambda(t)$, where $q_\lambda(t)$ is the concave component. Particularly, $p_\lambda(t)$ and $q_\lambda(t)$ satisfy the following regularity conditions:

1. There exists a constant $\nu$ such that the function $p_{\lambda}(t)$ satisfies $p'_{\lambda}(t) = 0, \forall~ t \geq \nu > 0$.

2. On the nonnegative real line, $q'_{\lambda}(t)$ is monotone and Lipschitz continuous, i.e., for $t' \geq t$, there exists a constant $\zeta_- \geq 0$ such that $ q'_{\lambda}(t') - q'_{\lambda}(t) \geq -\zeta_- (t'-t)$.

3. Both function $q_{\lambda}(t)$ and its derivative $q'_{\lambda}(t)$ pass through the origin, i.e., $q_{\lambda}(0) = q'_{\lambda}(0) = 0$.

4. On the nonnegative real line, $|q'_{\lambda}(t)|$ is upper bounded by $\lambda$, i.e., $|q'_{\lambda}(t)| \leq \lambda$.

For a matrix $\mathbf{M} \in \mathbb{R}^{d_1 \times d_2}$, define the nonconvex function $P_\lambda(\cdot)$ on the singular value vector of $\mathbf{M}$ such that $P_\lambda(\mathbf{M}) = \sum_{i = 1}^{d} p_\lambda(\sigma_i(\mathbf{M}))$, where $d = \min\{d_1, d_2\}$ and $\sigma_i(\mathbf{M})$ is the $i$-th singular value of $\mathbf{M}$.

Based on these definitions, I am wondering whether the following claim is correct or not that, for arbitrary $\mathbf{A, B} \in \mathbb{R}^{d_1 \times d_2}$, $$ P_\lambda( \mathbf{A + B}) \leq P_\lambda( \mathbf{A}) + P_\lambda( \mathbf{B}).$$

For a nonconvex function $p_\lambda(\cdot)$, with hyper-parameter $\lambda$, it can be decomposed into two parts that $p_\lambda(t) = \lambda |t| + q_\lambda(t)$, where $q_\lambda(t)$ is the concave component. Particularly, $p_\lambda(t)$ and $q_\lambda(t)$ satisfy the following regularity conditions:

1. There exists a constant $\nu$ such that the function $p_{\lambda}(t)$ satisfies $p'_{\lambda}(t) = 0, \forall~ t \geq \nu > 0$.

2. On the nonnegative real line, $q'_{\lambda}(t)$ is monotone and Lipschitz continuous, i.e., for $t' \geq t$, there exists a constant $\zeta_- \geq 0$ such that $ q'_{\lambda}(t') - q'_{\lambda}(t) \geq -\zeta_- (t'-t)$.

3. Both function $q_{\lambda}(t)$ and its derivative $q'_{\lambda}(t)$ pass through the origin, i.e., $q_{\lambda}(0) = q'_{\lambda}(0) = 0$.

4. On the nonnegative real line, $|q'_{\lambda}(t)|$ is upper bounded by $\lambda$, i.e., $|q'_{\lambda}(t)| \leq \lambda$.

For a matrix $\mathbf{M} \in \mathbb{R}^{d_1 \times d_2}$, define the nonconvex function $P_\lambda(\cdot)$ on the singular value of $\mathbf{M}$ such that $P_\lambda(\mathbf{M}) = \sum_{i = 1}^{d} p_\lambda(\sigma_i(\mathbf{M}))$, where $d = \min\{d_1, d_2\}$ and $\sigma_i(\mathbf{M})$ is the $i$-th singular value of $\mathbf{M}$.

Based on these definitions, I am wondering if the following claim is correct or not that, for arbitrary $\mathbf{A, B} \in \mathbb{R}^{d_1 \times d_2}$, $$ P_\lambda( \mathbf{A + B}) \leq P_\lambda( \mathbf{A}) + P_\lambda( \mathbf{B}).$$

For a nonconvex function $p_\lambda(\cdot)$, with hyper-parameter $\lambda$, it can be decomposed into two parts that $p_\lambda(t) = \lambda |t| + q_\lambda(t)$, where $q_\lambda(t)$ is the concave component. Particularly, $p_\lambda(t)$ and $q_\lambda(t)$ satisfy the following regularity conditions:

1. There exists a constant $\nu$ such that the function $p_{\lambda}(t)$ satisfies $p'_{\lambda}(t) = 0, \forall~ t \geq \nu > 0$.

2. On the nonnegative real line, $q'_{\lambda}(t)$ is monotone and Lipschitz continuous, i.e., for $t' \geq t$, there exists a constant $\zeta_- \geq 0$ such that $ q'_{\lambda}(t') - q'_{\lambda}(t) \geq -\zeta_- (t'-t)$.

3. Both function $q_{\lambda}(t)$ and its derivative $q'_{\lambda}(t)$ pass through the origin, i.e., $q_{\lambda}(0) = q'_{\lambda}(0) = 0$.

4. On the nonnegative real line, $|q'_{\lambda}(t)|$ is upper bounded by $\lambda$, i.e., $|q'_{\lambda}(t)| \leq \lambda$.

For a matrix $\mathbf{M} \in \mathbb{R}^{d_1 \times d_2}$, define the nonconvex function $P_\lambda(\cdot)$ on the singular value vector of $\mathbf{M}$ such that $P_\lambda(\mathbf{M}) = \sum_{i = 1}^{d} p_\lambda(\sigma_i(\mathbf{M}))$, where $d = \min\{d_1, d_2\}$ and $\sigma_i(\mathbf{M})$ is the $i$-th singular value of $\mathbf{M}$.

Based on these definitions, I am wondering if the following claim is correct or not that, for arbitrary $\mathbf{A, B} \in \mathbb{R}^{d_1 \times d_2}$, $$ P_\lambda( \mathbf{A + B}) \leq P_\lambda( \mathbf{A}) + P_\lambda( \mathbf{B}).$$