Is there hypothesis testing to describe "weak / no correlation"? In classical statistics literature, finding correlation between variables is important and we also have hypothesis testing on if correlation is 0 (example on this link)
I feel I need something opposite because my data is different. Unlike most "cleaned" data sets in statistics, my data is coming form many "sensors". There are a lot of measures/columns (say thousands of) and I do not know the details on what the sensors are measuring.
If I calculate correlation matrix (visualizing correlation matrix as an image), I would have such results (on a very small subset of data, think about I have a thousands by thousands matrix in real world). Where strong correlation is not interesting to me, because it shows the data is "redundant measure".

Therefore is there hypothesis testing to describe "weak / no correlation"? Or what are other metrics better for my case? Should I set a threshold on correlation metric and high light the pairs bellow the threshold? I am asking this question because I think it should have some formal way of doing this.

Thanks for whuber and DJohnson's comment and answer. I think I should put a quote on "correlation", since I am trying to describe the relationship between variables not Pearson correlation.
 A: Your question has several facets including understanding the structure of data that includes sparsity as well as clarifying what is meant by "correlation."
Xie and Xing have a nice paper Cauchy Principal Component Analysis which describes a typology (paradigm) of data structures based on whether they are probabilistic or non-probabilistic as well as their sensitivity to noise, denseness and sparsity of the information. They point out that classic Gaussian based PCA is "fragile to noise of large magnitude," and recommend alternative methods including their implementation of Cauchy PCA. 
I think you need to clarify what you mean by "correlation." A naive assumption is that you are referring to Pearson correlation, a standardized measure of linear association. A less naive and stringent assumption is that you could mean Spearman correlation for ordinal and monotonic association. Many other metrics of association are possible including nonlinear measures of dependence such as maximal correlations, others leveraging reduced kernel Hilbert space (RKHS) such as distance correlations, partial distance correlations, and information theoretic, entropy-based measures such as the Reshef brothers' mutual information criterion (MIC). 
Once your meaning of the term "correlation" is clarified, then people can begin to address your concerns with magnitudes of association.
A: Larry Wasserman discusses building graphical models from hypothesis tests on the the correlation between variables in his stats course: 
http://www.stat.cmu.edu/~larry/=stat705/
There's a nice video of his lecture on the class page and the lecture notes are here:
http://www.stat.cmu.edu/~larry/=sml/GraphicalModels.pdf
